Algebra

Table of Contents

Notation

  • algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png denotes a group
  • CANI stands for:
    • Commutative
    • Associative
    • Neutral element (e.g. 0 for addition)
    • Inverse

Terminology

almost all
is an abbreviation meaning "all but finitely many"

Definitions

Homomorphisms

A homomorphism algebra_849512a42edfa45960f881a7df646707be178c8a.png is a structure-preserving map between the groups algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png and algebra_510a3d993dd428bfe79de35a9511a7ed4f30995e.png, i.e.

algebra_4165f6f7cd1486b06dcaf5ddd68c05f6bc9ae34b.png

A endomorphism is a homomorphism from the group algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png to itself, i.e. algebra_dea6fb3d8acb9b069aa69dab2566725344114587.png

An automorphism is an invertible homomorphism, denoted algebra_910ee3c016794ce25a8e02fa44ffcfbf7f9e581b.png.

In the case of a vector space algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png, we have

algebra_9dc7550141c9f9e7347c7e0e957bd25541f4e64e.png

Linear isomorphism

Two vector spaces algebra_f377695d881c4f672d1bf48b016adc1c0890005c.png and algebra_0cc551b48f3090ef07007428021e3f7fa971cdcf.png are said to be isomorphic if and only if there exists a linear bijection algebra_ba29ecd28cc957aa0130175d00942161ae02f4a2.png.

Field

An (algebraic) field algebra_5b269ac8577606f24d0743f409e32346d697203a.png is a set algebra_ab9d9977871eda0af750024aad11a3b9b2f6c240.png and the maps

  • algebra_9f838f9b1f91354ba346d6484ef0aef828a89dd0.png
  • algebra_bc701cd44eb3f15187edcca1aa17ad203d04cd3e.png

that satisfy

algebra_7511b29800dbe70b8d245b9f12daf6dc8572d514.png

Vector space

A vector space algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png over a field algebra_d5c60a4ae2b31ef1ab0bdd9fc92339f09358331b.png is a pair consisting of an abelian group algebra_044b29cb3e6e5e44e0508d5ece0b1140d63c0ffc.png and a mapping

algebra_7d923630eca01f520a745c37a1182939c013dddc.png

such that for all algebra_1a4bb9aa1930921b069a67c25332c0a9c9caf24f.png and algebra_93d5c1cf6271bf22ba2f53da7da1cf60e6a238e0.png we have A D D U:

  • Associativity: algebra_ea345f6e58ba37c6dbc95d94a3828d636d8d4d60.png
  • Distributivity over field-addition: algebra_95f09179658db2a21634a99c1e5483f9cf1cd151.png
  • Distributivity over field-multiplication: algebra_63e65f9f4c0714a73549330a63ad07f12c018687.png
  • Uint: algebra_72427b49a98e76d5f46aaa6d8cfd5541405e2902.png

Ring

A ring algebra_c32ad2bd4c865b2990252fb1ee7894b8346d96a8.png over a set algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png with the maps:

  • algebra_9f838f9b1f91354ba346d6484ef0aef828a89dd0.png
  • algebra_bc701cd44eb3f15187edcca1aa17ad203d04cd3e.png

which satisfy the same as a field, but without C and I for algebra_c73ba8d9fee3e8c29911481fc0726828c3549c28.png, i.e. multiplication.

A division ring is a ring where the multiplicative operation algebra_c73ba8d9fee3e8c29911481fc0726828c3549c28.png allows an inverse, i.e. it's

algebra_887794784b2da6e4f3c90cd7a07579a923e37cac.png

Equivalence relations

A equivalence relation on some set is defined as some relation betweeen algebra_cb254ffe3e498884a0e8a2679f6c3a003f7219a5.png and algebra_62d0596bf7b968d41aa765c30eee0fa137f87c97.png, denoted algebra_2e2ff4d83614dcad6e119757d6fcc505eba4d0d1.png, such that the relation is:

  • reflexive: algebra_a092660975a13b25ce2a0f06977c5f46f52b8ccc.png
  • symmetric: algebra_dbc8e8f1eb109853aa528b65ae67394fb2313a85.png
  • transistive algebra_3e5710dc796803e241d492754a44abd9fe9cb34b.png

Why do we care about these?

  • Partitions the set it's defined on into unique and disjoint subsets

Unitary transformations

A unitary transformation is a transformation which preserves the inner product.

More precisely, a unitary transformation is an isomorphism between two Hilbert spaces.

Groups

Notation

  • algebra_808dc4aafa8b1616ebfd1d8f09a625b5d84810b5.png or algebra_f110cf528bd1fdde925934898ef4845abf56e949.png denotes the number of left cosets of a subgroupd algebra_510a3d993dd428bfe79de35a9511a7ed4f30995e.png of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png, and is called the index

Definitions

The symmetric group algebra_9d0a0b11e7edee28a295165fcb74c504242d686a.png of a finite set of algebra_5403f6bbface4889f05450c96efe5dfafd041d71.png symbols is the group whose elements are all the permutation operations that can be performed on the algebra_5403f6bbface4889f05450c96efe5dfafd041d71.png distinct symbols, and whose group operation is the composition of such permutation operations, which are defined as bijective functions from the set of symbols to itself.

An action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of the space.

If algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png is a group and algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png is a set, then a (left) group action algebra_a14d8487fae6909c522cdd676b01387e7c5d213d.png of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png on algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png is a function

algebra_638d75def99d9e3873530e6f276bb93b5d0e773b.png

that satisfies the following two axioms (where we denote algebra_b10893cc554b3d012c48038642559621910f3a29.png as algebra_7d7c456dc1d2be1c7fdb0f4649b830734289f939.png):

  • identity: algebra_13daf44fa7341689cb431d28ac11692f39fbee36.png for all algebra_46bb2460b54adf6ecdb59225d2987f80531ea4cc.png (algebra_2d4303418f92a7b5e2a7f60f86adbd3ba85fc665.png denotes the identity element of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png)
  • compatibility: algebra_72dd9d5bfa0259199eb076581f890b920ce47ea0.png for all algebra_fee9c179246b858a944ff3e70d495ac7473402f3.png and all algebra_46bb2460b54adf6ecdb59225d2987f80531ea4cc.png

where algebra_8e58a77f62ed0b8051c2efee34e172dba8073e37.png denotes the result of first applying algebra_c4207654f80c68e96d536088ad629a9dfa3c6927.png to algebra_2a61cee9c707231c73a1fcb6992bc4bb552044fe.png and then applying algebra_c4f480233088a134e88f2426541b2f00ca318b55.png to the result.

From these two axioms, it follows that for every algebra_74e4e93e9fe137f40a50a0770f6879afdfd86603.png, the function algebra_a14d8487fae6909c522cdd676b01387e7c5d213d.png which maps algebra_46bb2460b54adf6ecdb59225d2987f80531ea4cc.png to algebra_053cca061d2fa8bf6f81ed01328a0f29ed109d7c.png is a bijective map from algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png to algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png. Therefore, one may alternatively define a group action of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png on algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png as a group homomorphism from algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png into the symmetric group algebra_1e3edd37fe7d60878e60a0bdac4e8b7431140a97.png of all bijections from algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png to algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png.

The action of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png on algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png is called transistive if algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png is non-empty and if:

algebra_4a5828fc8d66086675afccd7321a5981495eb756.png

Faithful (or effective ) if

algebra_904fbe05c00856a3e7f22d903ff64c65aa30f106.png

That is, in a faithful group action, different elements of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png induce different permutations of algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png.

In algebraic terms, a group algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png acts faithfully on algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png if and only if the corresponding homomorphism to the symmetric group, algebra_142510ca5e2088cab8c0a08bd3656e352acfe8bb.png, has a trivial kernel.

If algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png does not act faithfully on algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png, one can easily modify the group to obtain a faithful action. If we define:

algebra_56f5762ceaf4343cf8d35c300561bda7514372f1.png

then algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png is the normal subgroup of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png; indeed, it is the kernel of the homomorphism algebra_142510ca5e2088cab8c0a08bd3656e352acfe8bb.png. The factor group algebra_eac76664013e2fcbc79fd930b69659104b99c83e.png acts faithfully on algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png by setting algebra_6d42a5daf91f6509c2466c046f805ba34a162311.png.

We say a group action is free (or semiregular or fixed point free ) if, given algebra_fee9c179246b858a944ff3e70d495ac7473402f3.png,

algebra_69c5f47a92f4541addabfea53d33c7937b0b49b2.png

We say a group action is regular if and only if it's both transitive and free; that is equivalent to saying that for every two algebra_798d8d1a454ea9b8a4f335042f69838743b14e9e.png there exists precisely one algebra_74e4e93e9fe137f40a50a0770f6879afdfd86603.png s.t. algebra_960167312553a00a320efb490a2138196d5ce0c6.png.

Consider a group algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png acting on a set algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png. The orbit of an element algebra_2a61cee9c707231c73a1fcb6992bc4bb552044fe.png in algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png is the set of elements in algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png to which algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png can be moved by the elements of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png. The orbit of algebra_2a61cee9c707231c73a1fcb6992bc4bb552044fe.png is denoted by algebra_ad2d983c3e87dd0abf74d1ca85438c36b438db5d.png:

algebra_8597b73efe36bd543370b64e15da8712a953d540.png

An abelian group, or commutative group, is simply a group where the group operation is commutative!

A monoid is an algebraic structure with a single associative binary operation and an identity element, i.e. it's a semi-group with a binary operation.

Cosets

Let algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png be a group and algebra_510a3d993dd428bfe79de35a9511a7ed4f30995e.png be a subgroup of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png. Let algebra_74e4e93e9fe137f40a50a0770f6879afdfd86603.png. The set

algebra_1e3bedf37f589f979f13f1a0ca3a5772be864bb0.png

of products of algebra_c4f480233088a134e88f2426541b2f00ca318b55.png with elements of algebra_510a3d993dd428bfe79de35a9511a7ed4f30995e.png, with algebra_c4f480233088a134e88f2426541b2f00ca318b55.png on the left is called a left coset of algebra_510a3d993dd428bfe79de35a9511a7ed4f30995e.png in algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png.

The number of left cosets of a subgroup algebra_510a3d993dd428bfe79de35a9511a7ed4f30995e.png of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png is the index of algebra_510a3d993dd428bfe79de35a9511a7ed4f30995e.png in algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png and is denoted by algebra_d54bb6b06ba527cdb702a85f1c7d87b6c773c748.png or algebra_808dc4aafa8b1616ebfd1d8f09a625b5d84810b5.png. That is,

algebra_80d41f30e681ceec321ae5d09859132c2a8d5b67.png

Center

Given a group algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png, the center of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png, denoted algebra_5448af5aaaddf6ec7fb281696d68c516c625f4f6.png is defined as the set of elements which commute with every element of the group, i.e.

algebra_cc3789299356be29ed9485e6afaa5f713c4a2b5d.png

We say that a subgroup algebra_510a3d993dd428bfe79de35a9511a7ed4f30995e.png of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png is central if it lies inside algebra_5448af5aaaddf6ec7fb281696d68c516c625f4f6.png, i.e. algebra_7d15efc985b7729709eb746c035a4bebfd9362d1.png.

Abelianisation

Given a group algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png, define a abelianisation of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png to be the quotient group

algebra_0a59fbe199b34ac0a0e62ee1d2b15ca1a1e533f3.png

with algebra_7387cc6f2f604b122061f017d83448e6c6b9c884.png is the normal subgroup generated by the commutators algebra_89c7e48ab52999e84ad17393175fb3e19c343abc.png for algebra_92ebeab2ae7c9c48db591e4a712157a00db05b19.png, i.e.

algebra_da156ab3fd3fae6c06b169b6c75106b449440d18.png

Theorems

Let algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png be a group of order algebra_5403f6bbface4889f05450c96efe5dfafd041d71.png and let algebra_b346b8eaf5913ed81b033dacc1d6568c142bbb4a.png be a prime divison of algebra_5403f6bbface4889f05450c96efe5dfafd041d71.png.

Then algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png has an element of order algebra_b346b8eaf5913ed81b033dacc1d6568c142bbb4a.png.

Fermat's Little Theorem

Let algebra_b346b8eaf5913ed81b033dacc1d6568c142bbb4a.png be a prime number.

Then for algebra_f97d0230e02c277a4d0b4604a17bcb5063885083.png, the number algebra_7c96dfa9c1ebd915cef30157fe53a774fea6ba53.png is an integer multiple of algebra_b346b8eaf5913ed81b033dacc1d6568c142bbb4a.png. That is

algebra_bdeb7da937082e4cdea022af3c488ec3c0673e84.png

Isomorphism theorems

Let

Then algebra_36eda277e3798c1bdda254ec79ff530e2fc35fa1.png is a normal subgroup of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png, and algebra_7a1e4fc2b5eb0f040686369beff84f8e11a33473.png. Furthermore, there is an isomorphim

algebra_09f98de697d8ea7de68064f4cac145187a92803c.png

algebra_12a6923764150bc68831454b205d39c0e38a2ec3.png

In particular, if algebra_a14d8487fae6909c522cdd676b01387e7c5d213d.png is surjective, then

algebra_05d15785197a732eb0f01e6f16987af50bde12eb.png

Le algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png be a group, algebra_2482305b4aa70cf34b4dd95c8e7e263106369868.png and algebra_dd019968d1905aedd05462c538ec157c47bf58c4.png.

  1. algebra_ae25a2395008d8c5c7a795cd4e199757267c8c17.png
  2. algebra_f8141d10c341f05369bb2ed83e77b65d96895e2a.png
  3. algebra_1aa3f18c5b8a6c245bafaf0ea6a617fd8803e753.png
  4. algebra_a66e7790f6f0cf43efd05d0d65cff73e8d2a98f9.png
  1. First algebra_eb8ba9f70a4df6d288f436ac697af49d50856980.png so clearly algebra_098e475c7560285f79eac92b11719fa23f974a55.png. Let algebra_240401863411856be069c90492cc5bd6a2f16099.png and algebra_a35636635eddd903c1015d3ab2cff21d29d8a9cc.png. We then want to show that algebra_243b08379bfe7a14162ad8f9ad7d3e0a78ce6eb3.png. Observe that

    algebra_e5172c14a345997dba863bf2d8bd999eae20887d.png

    since algebra_bd4a069d2c8b2dedf515030be1252f1c77534aa9.png. So algebra_466955ad0f713c3f522ce4b249d55957b8c7f11f.png so that algebra_a9183f03a2d0a210e137b7e0c54f59ab677e0c88.png,

    algebra_ed8ce3e8a3244a796a85674eab67952c503bd15c.png

    And now we check that the algebra_60c10ce277a28373d46f059ef41c91d24f61ae8a.png

    algebra_e0bab1d1a8b13b3d0d3ac11ec732b7029529332d.png

  2. Let algebra_2c5beaf23a21016b8d5aa7616e3eb2e28e912b7f.png, algebra_e474354dfc4d611d3e0308b0a9492722117ac7e9.png. Wan to show that algebra_755d3991e89157b8fdbdf6275c69963ebcf3ff2d.png.

    algebra_94e49fc9de0e0693faacad1ceee6b88311202b85.png

  3. Let algebra_3bf653a667840d4fa9a3f0c297f71a1275e5040e.png, algebra_7d1e190808a6248d454279b21605f32e0d37669f.png. Then algebra_2956a85a7c062ef2952a960e339524de0e0bb0fd.png since algebra_2482305b4aa70cf34b4dd95c8e7e263106369868.png, so algebra_2956a85a7c062ef2952a960e339524de0e0bb0fd.png, for all algebra_74e4e93e9fe137f40a50a0770f6879afdfd86603.png.
  4. Want to show that algebra_efeaa9883bb563ab6487ddafa855f29d035ea659.png. First Isomorphism Theorem tells us that

    algebra_603a196bb8256aa3977258900883dcb09289ea2b.png

    Therefore, letting

    algebra_97d1de2e62fb134b14855405c94e4bfebb03486b.png

    where we simply factor out the algebra_5403f6bbface4889f05450c96efe5dfafd041d71.png from every element in algebra_e54640c3438f73c8ef89d6c6551133ef0e738a35.png.

    algebra_ced2764f895154b14773d52e39d7f670094c65b5.png

    So algebra_a14d8487fae6909c522cdd676b01387e7c5d213d.png maps into algebra_53f0515ba4668f6b21a92644afe9083f1849704a.png, but we need it to be surjective, i.e.

    algebra_932f331934b71076f04303e23511a1269b7d6429.png

    An element of algebra_53f0515ba4668f6b21a92644afe9083f1849704a.png is a coset algebra_1f3c2cd61bb0d0b93b03bd96dbc033c4e7058db7.png for algebra_aa4d453ede50352c3b6c55859d27ed94d5ff7e1b.png, which is clearly algebra_0e44f40e85a848d05711922bfe94f7ba4d2dd961.png. And finally,

    algebra_1e88d591c80886d9d7434debbcd60cc89f77ac42.png

    Hence, by the First Isomorphism Theorem,

    algebra_74cc704fdca853f0473a5b5f2300982159534ec5.png

Notice algebra_81ce549ca5b626d41942cc363089f17c9be68e0f.png. In fact, algebra_db34b66627fa598f470153015bb8993de920445b.png.

As an example, algebra_869f02da3763928d34a7df56fdf965798dc57309.png. Then

algebra_f6c45bc2499c7aaa005e9f272613b320c47b8c8d.png

by the First Isomorphism Theorem and since algebra_52523edf96eaa18172f6bceff6a2c12206bf0fde.png. And, we can also write

algebra_02d4410bd6792ac984c980e19a3ed3f15be43660.png

since algebra_69e3151218472e58d542004cefce35757c3eb15a.png.

Want to show that both are isom. to algebra_5bb285e6902234d029c168e57f18f1da4f2562af.png. We do this by constructing map:

algebra_e844b78120fb1bb1cdd914ab53d6a3a853048f00.png

and just take the coset. And,

algebra_121acafacecde7a564d76dd9f81e167d6b877f6b.png

Hence by 1st Isom. Thm.

algebra_1acacb5548b907d782fcedaa3e7b58a6ad09b87c.png

Kernels and Normal subgroups

Arising from an action algebra_c5fa39e4efed04922236a1cf52d4925c5a6011e6.png of a group algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png on a set algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png, is the homomorphism

algebra_03c28732b11203105b5b141e7d066b522999d1bd.png

the permutation of algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png corresponding to algebra_c4f480233088a134e88f2426541b2f00ca318b55.png.

The kernel of this homomorphism is also called the kernel of the action algebra_c5fa39e4efed04922236a1cf52d4925c5a6011e6.png and is denoted algebra_424688da2a65e705e02a182e5e7d5d279e1c3721.png.

Remember,

algebra_656e3b7961fcaef20514f38473e54213b278cf97.png

is the permutation algebra_3c09f85b5698972ec8c5152d2a526e77d000ac38.png. Therefore, algebra_a2c5d6a845645796bfc3dc06e01b59b010621886.png if and only if algebra_393fa0fdb4896c45b00fea968723506a1106d0fa.png for all algebra_46bb2460b54adf6ecdb59225d2987f80531ea4cc.png, and so

algebra_d5035c19f84654db3fe2b02bcb790015f819fba1.png

consists of those elements which stabilize every element of algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png.

Let algebra_510a3d993dd428bfe79de35a9511a7ed4f30995e.png be a subgroup of a group algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png, and let algebra_f5fd7b63d088d56208bb47bbb9a1b08113aaccb6.png.

The conjugate of algebra_510a3d993dd428bfe79de35a9511a7ed4f30995e.png by algebra_cb254ffe3e498884a0e8a2679f6c3a003f7219a5.png, written algebra_b960b298e130b7c9b797f4e74b4d7e763ddcb7c8.png, is the set

algebra_eb736a35764824cadf9cda914021e4f378bc407b.png

of all conjugate of elements of algebra_510a3d993dd428bfe79de35a9511a7ed4f30995e.png by algebra_cb254ffe3e498884a0e8a2679f6c3a003f7219a5.png.

This is the image algebra_b0e6fe00c973bbbf964686edbb4606e5267705bf.png of algebra_510a3d993dd428bfe79de35a9511a7ed4f30995e.png under the conjugation homomorphism algebra_cbe81e69ed9b8eed5c104b200cfa014bbd6d5562.png where algebra_aa9145610066372bfc562bab02488234047ce3b4.png.

Hence, algebra_b960b298e130b7c9b797f4e74b4d7e763ddcb7c8.png is a subgroup of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png.

Let algebra_510a3d993dd428bfe79de35a9511a7ed4f30995e.png be a subgroup of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png.

If algebra_e71c4c73e2e32b26d5b6613803059eed4b84509f.png for all algebra_f5fd7b63d088d56208bb47bbb9a1b08113aaccb6.png, then

algebra_d180d2862fa3e5d3f13323e84e573ff6f71f1c45.png

A subgroup algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png of a group, algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png, is called a normal subgroup if it is invariant under conjugation; that is, the conjugation of an element of algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png by an element of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png is still in algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png:

algebra_d22ebd66cd269f26ae05574bfa604704ecdde0e1.png

algebra_cd69b57821033a3237a428de4eae4d1acd51589d.png. Then algebra_af4f36c2644234163fd4d4bc717455feff8d6e0f.png if and only if algebra_30060eb92a6cfe5ca612b2da82053285542e78b2.png for all algebra_74e4e93e9fe137f40a50a0770f6879afdfd86603.png.

Let algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png and algebra_510a3d993dd428bfe79de35a9511a7ed4f30995e.png be groups.

The kernel of any homomorphism algebra_d4fc9fb418773f4dff6cd15b2e0d2f8ec95507d8.png is normal in algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png.

Hence, the kernel of any group action of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png is normal in algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png.

Let algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png be a group acting on the set algebra_02e438523616b97120e7df10937e0cbb6b965aa8.png of left cosets of a subgroup algebra_510a3d993dd428bfe79de35a9511a7ed4f30995e.png of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png.

  1. The stabilizer of each left coset algebra_139aa0d499555ddf0027ccd21f2df4722d4c115e.png is the conjugate algebra_7fb3ee11380c3e0a1a301c7bd427f445320c20f4.png
  2. If algebra_1c96ae8fe5ba1fa1a2817ea059203a1cef79e2c1.png for all algebra_f5fd7b63d088d56208bb47bbb9a1b08113aaccb6.png then algebra_46938ace78d4fc9f3a74876ef4b1116e34b146c7.png

Factor / Quotient groups

Let algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png be a group acting on a set algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png and algebra_ab9d9977871eda0af750024aad11a3b9b2f6c240.png be the kernel of the action.

The set of cosets of algebra_ab9d9977871eda0af750024aad11a3b9b2f6c240.png in algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png is a group with binary operation

algebra_5506f7f7cfcf2f625355ed535702b4b2730717af.png

which defines the factor group or quotient group of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png by algebra_ab9d9977871eda0af750024aad11a3b9b2f6c240.png and is denoted

algebra_2cdcfc084a3a2d1c721f1dbd060778665f05332d.png

But I find the following way of defining a quotient group more "understandable":

Let algebra_de6dee8ff24542df430e2c09d7beb7154c1c9424.png be a group homomorphism such that

algebra_99df45dc3c4efd789adfbd0f26f94bd604529a08.png

That is, algebra_a14d8487fae6909c522cdd676b01387e7c5d213d.png maps all distinct algebra_6bb98af50d6e3adf9175ae0149cae3ce49b3b832.png which are equivalent under algebra_a2c680518aeb36775cf4f8b9613d6507a85e0690.png to the same element in algebra_da7be4dce5954506206e5d807c4fe1ed49551fbb.png, but still preserves the group structure by being a homomorphism of groups.

There is a function algebra_c6bbf81936453826dd4994744f7a53bb51f50475.png with

algebra_eba7b264c74e040bf6592b9e4f900a10c9f070be.png

For algebra_92ebeab2ae7c9c48db591e4a712157a00db05b19.png we have that

algebra_145387d7f64c32e128079cecd217f0ceb7fb2d59.png

thus algebra_f017a9b0f9e8a176a3db97e0116f98ff496ac318.png is a homomorphism.

Then, clearly

algebra_4a3ade98b1ebe367184510bdf6ec2b3b17c2c303.png

algebra_f017a9b0f9e8a176a3db97e0116f98ff496ac318.png is called the natural homomorphism from algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png to algebra_eac76664013e2fcbc79fd930b69659104b99c83e.png.

The order of a factor group algebra_eac76664013e2fcbc79fd930b69659104b99c83e.png is the number of distinct cosets of algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png in algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png.

If algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png is finite, then

algebra_ac171997390ee3272d5de20c80cd7712eb183dd8.png

Group presentations

Notation

  • algebra_43b335c5c3ae62be10d913b117b1004525671995.png or algebra_ed47d745f8c05a56cdc26e08386de88e10d80146.png refers to the group generated by algebra_2a61cee9c707231c73a1fcb6992bc4bb552044fe.png such that algebra_84858d6df4a436fea9ecd932527942be1647f8cd.png
  • algebra_bbf8be405bf0d7b1222410b34725d65e5b891e54.png denote is the free group as generated by algebra_2a61cee9c707231c73a1fcb6992bc4bb552044fe.png
  • In general: algebra_36efdb81c5b748f2a554c6de2ee41bf343cf046a.png, e.g.

    algebra_3abd428dfb5de312c3bc85e42747d92ddbdc29e2.png

    where the "unit-condition" simply specifies that the group is commutative

Free groups

algebra_f1b731ec5f22407a74afed17dab058f477a72734.png is the free group generated by algebra_12e7e25784c70de09d0d6151cf9536c80def11d6.png.

Elements are symbols in algebra_5a54c079e486f61ab494e0775f470e3fa797d212.png, subject to

  • group axioms
  • " and all logical consequences :) "

Let algebra_6591b7b5415045ca462e77578ba89ac47ad01eb0.png.

The group with presentation

algebra_cec8aa75ad271aa353637d281e26cce4a273ad58.png

is the group generated by algebra_12e7e25784c70de09d0d6151cf9536c80def11d6.png subject to

  • group axioms
  • algebra_e69e862a25b8f02ba87cfecf5502f829a4a0c158.png
  • " and all logical consequences :) "

There's no algorithm for deciding whether algebra_9ccaa8f1bc7bf60d2219c0c8e9689b338ed76b1c.png is the trivial group.

Let algebra_e300b14701bbccc0a81b705d8a1d5b0b4bfdcea7.png and let algebra_d5bef3600337acb4eb09a951d399184e95d1fd50.png, where algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png is a group.

Then there is a unique homomorphism:

algebra_acf13ada0baf5335152c2572a25f9cf0b35a0292.png

And the image of algebra_108e48865692dff7955ba9ddbee0c1e718d23960.png is the subgroup of algebra_8ad47b33d9db4dafb715f077ba205eeab9cecff8.png generated by algebra_085106eacb32695e632f6a1b6b5ef1dfe74efd88.png.

Exact sequence

An exact sequence of groups is given by

algebra_6cadc302fa857e12b573c75369df34be7d0d3b71.png

of groups and group homomorphisms, where exact refers to the fact that

algebra_7276f742f4e6b1c3f6befcaf9e96ef813aed91dc.png

Linear Algebra

Notation

  • algebra_b04ca1b2a60559bff978fe2a1152d21bf4cc6185.png denotes the set of all matrices on the field algebra_d5c60a4ae2b31ef1ab0bdd9fc92339f09358331b.png
  • algebra_f3040a1696f55797890ecd078452fa73b0f7afa9.png denotes the representing matrix of the mapping algebra_b314324651ef278db247a6282c2c7967d1e6c28d.png wrt. bases algebra_4cc68ebefc13f65a9f085f468fdf9abda339a257.png and algebra_49af2ec32d071a93d42ea9f6cba4ef37624c1e14.png, where algebra_4cc68ebefc13f65a9f085f468fdf9abda339a257.png is ordered basis for algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png and algebra_49af2ec32d071a93d42ea9f6cba4ef37624c1e14.png ordered basis for algebra_3bd1f26e899259191c15988046ebcac426ab307c.png:

    algebra_be85454a47d13e202b5c4350316bd43be7a76ac1.png

  • algebra_94a12e1b034df7a0e0b961b4e0786048372857df.png and algebra_9869d539bd2f186ad9d73d080a4c7cbf550756af.png where algebra_48f5b9e38dd578d7a95805a88eb7bde12c31fca4.png denotes the "identity-mapping" from elements represented in the basis algebra_4cc68ebefc13f65a9f085f468fdf9abda339a257.png to the representation in algebra_49af2ec32d071a93d42ea9f6cba4ef37624c1e14.png.
  • algebra_ba347c28188e58ffef17d601abfbe6be4f4cc337.png denotes the n-dimensional standard basis
  • algebra_8d4dcb61bd7b7486d3a66a91fc734ae61335fafc.png
  • algebra_69c85388a47a995be9d6174d887100ee4c835fa1.png, i.e. the set of non-zero elements of algebra_d5c60a4ae2b31ef1ab0bdd9fc92339f09358331b.png

Vector Spaces

Notation

  • algebra_f40ad5f32532ae52dd17a4315b7711042277a778.png is a set
  • algebra_d5c60a4ae2b31ef1ab0bdd9fc92339f09358331b.png is a field
  • algebra_a2413718a527a7ecfac365da653d8ac2fbebd683.png
  • algebra_84aa56b11ef39ba32d535d3bd57a4b12c910f4be.png

Basis

A subset of a vector space is called a generating set of the vector space if its span is all of the vector space.

A vector space that has a finite generating set is said to be finitely generated.

algebra_2a434caf22d3292fec4d8a3f5578a0562f0776bf.png is called linearly independent if for all pairwise different vectors algebra_fb19639ae983ef42b079e4d7f189d3049cd0a243.png and arbitrary scalars algebra_37affb82d0d5d2712a51f98254b29947d53974f0.png,

algebra_54ab0e90730b764278e4a44f767b042ba67e772d.png

A basis of a vector space algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png is a linearly independent generating set in algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png.

The following are equivalent for a subset of algebra_60a08d4539c1360eb539b1e33b4a20167f7490b5.png for a bector space algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png:

  1. algebra_a1e2c57ce3a3d2af123967adb15ba31a234a00fc.png is a basis, i.e. linearly independent generating set
  2. algebra_a1e2c57ce3a3d2af123967adb15ba31a234a00fc.png is minimal among all generating sets, i.e. algebra_63552425cfe7034c44b15b0f7dc60feaf105c4f3.png does not generate algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png for any algebra_1c54fba801ca72d4a49209e99dfaea752286cbfa.png
  3. algebra_a1e2c57ce3a3d2af123967adb15ba31a234a00fc.png is maximal among all linearly independent subsets, i.e. algebra_524f105384a261ec356801f48412051213552500.png is not lineraly independent for any algebra_964f5c0895be0566e8f3bc7cc6084dab9ed9164f.png.

"Minimal" and "maximal" refers to the inclusion and exclusion.

Let algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png be a vector space containing vector subspaces algebra_29ae74555c01fc8b37cbe123c74f79f0b82e55ba.png. Then

algebra_70cf84ab1243bf77517b76e41661b7c5372fcbab.png

Linear mappings

Let algebra_07eb6a6d51bab611982313d706c4993ed7dd9d45.png be vector spaces over a field algebra_d5c60a4ae2b31ef1ab0bdd9fc92339f09358331b.png. A mapping algebra_b314324651ef278db247a6282c2c7967d1e6c28d.png is called linear or more precisely algebra_d5c60a4ae2b31ef1ab0bdd9fc92339f09358331b.png linear if

algebra_4586ad1662cfbb8feab98de0960fe0bc50df46eb.png

This is also a homomorphism of vector spaces.

Let algebra_32296dadd7b7102ed7eaf73a6984bdf6884e5729.png be a linear mapping between vector spaces. Then,

algebra_985b9f037b0af0864583c8cac7053b732fa908e3.png

where usually we use the terminology:

  • rank of algebra_f017a9b0f9e8a176a3db97e0116f98ff496ac318.png is algebra_fa1b86ecadb7d85678ad9b88e2efd97a93918885.png
  • nullity of algebra_f017a9b0f9e8a176a3db97e0116f98ff496ac318.png is algebra_71f21b6bf803d42eefd5beb84f02bffbb049c0c2.png

Linear Mappings and Matrices

Let algebra_d5c60a4ae2b31ef1ab0bdd9fc92339f09358331b.png be a field and let algebra_cd3f18541158b8489c957e9d783988a89bb52cb9.png.

There exists a bijection algebra_16ab73d77396735a213529c541caa3b460d155a2.png and set of matrices with algebra_5403f6bbface4889f05450c96efe5dfafd041d71.png rows and algebra_8f44de754519a6b4b737f2e24c2083a1a7cb4a03.png columns:

algebra_434cd7340933f8f056c4c4e43333f237220b73f5.png

Which attaches to each linear mapping algebra_f017a9b0f9e8a176a3db97e0116f98ff496ac318.png, its representing matrix algebra_df55782e1c5eecc60d1d21d53f343ca639d2d425.png, defined

algebra_5628eabce1b4eb70002e9f9602b290ac1784456b.png

i.e. the matrix-representation of algebra_f017a9b0f9e8a176a3db97e0116f98ff496ac318.png is a defined by how algebra_f017a9b0f9e8a176a3db97e0116f98ff496ac318.png maps the basis of the target space.

Observe that the matrix product between two matrices algebra_8dbb167029e53bb938c37d3987f0c6e58c71dc2c.png and algebra_44427fa947e041f12421a804aab07ff7a9aa4ca0.png,

algebra_c1c373773af339cdebfc2a4076cf2c6676875b78.png

An elementary matrix is any square matrix which differs from the identity matrix in at most one entry.

Any matrix whose only non-zero entries lie on the diagonal, and which has the first 1's along the diagonal and then 0's elsewhere, is said to be in Smith Normal Form

algebra_823ecd90121ced534bcd932273a8e471a09e1adc.png

algebra_6bead628ff451cb3643691493efc410109bbcfc4.png there exists invertible matrices algebra_a0e42a3969013d94a065630de2ab62bb3d124b8f.png and algebra_2fb79be669b0dd19314e865fa5fac86d032cd33d.png s.t. algebra_11e8be6fb4b23a75a6279d9dfa9d8a71991dba34.png is a matrix in Smith Normal Form.

A linear mapping algebra_153a7805d374f325cb3056fa08e3d9ff90deb8c6.png is injective if and only if

algebra_3fa5feee364089006dca69f467343be84453b924.png

Let algebra_93d5c1cf6271bf22ba2f53da7da1cf60e6a238e0.png, then

algebra_0a57a548412babbb45f5622deaf52fd7ff2a8f01.png

Hence, if

algebra_db91f95c5a69d674020071395f3ed14eee7b8d85.png

as claimed.

Let algebra_64f80468940fcef37d9516ec2ced08289344cde2.png be square matrices over some commutative ring algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png are conjugate if

algebra_83b015de0c3031c57ccb88e3f9fcd845cdad5ebb.png

for an invertible P ∈ (n; R).

Further, conjugacy is an equivalence relation on algebra_e1347674c5193673b1f42833898595604a5d0459.png.

Trace of linear map

The trace of a matrix algebra_f75be59687ef2da76b04e88af02585ab68efc56c.png is defined

algebra_b860e2ad082e94067c00b03fbe027c582e39cac2.png

The trace of a finite product of matrices algebra_d4c59121bd7d2439a1bc032407570d1ab53676c6.png is independent of the order of the product (given that the products are valid). In other words, trace is invariant under cyclic permutations.

To see that trace is a invariant under cyclic permutations, we observe

algebra_f732101df6256a6ea1d9e592379db8e0c11ff44d.png

This case of two matrices can easily be generalized to case of products of multiple matrices.

Rings and modules

A ring is a set with two operatiors algebra_dc84ec6d92aa917bdf437d430d34787c1cc32a79.png that satisfy:

  1. algebra_3028dcb9ad64ce85f9227a2e7990eff72f7fc8f8.png is an abelian group
  2. algebra_0fabca862f164a0ee2a96fc84b9a8527429e0e1a.png is a monoid
  3. The distributive laws hold, meaning that algebra_b6afcabd4a0bd7c5320552aff5afedcc710b5034.png:

    algebra_727b55f26a598aa96a1ae21c8f57df84d200ae01.png

Important: in some places, e.g. earlier in your notes, they use a slightly less restrictive definition of a ring, and in that case we'd call this definition a unitary ring.

Polynomials

A field algebra_d5c60a4ae2b31ef1ab0bdd9fc92339f09358331b.png is algebraically closed if each non-constant polynomial algebra_1e73f69d403bf36e7be23ca733e5f3472a80ac30.png with coefficients in algebra_d5c60a4ae2b31ef1ab0bdd9fc92339f09358331b.png has a root in algebra_d5c60a4ae2b31ef1ab0bdd9fc92339f09358331b.png.

E.g. algebra_8c72062dd06f8f31b1004fd35828c6fca2f4ec24.png is algebraically closed, while algebra_3caab67ae7e6823d845012c0c5fc7b0b3c14d980.png is not.

If a field algebra_d5c60a4ae2b31ef1ab0bdd9fc92339f09358331b.png is algebraically closed, then every non-zero polynomial algebra_90f5e10aa3db1147f97d3417323d7a298d3ea617.png decomposes into linear factors

algebra_1bbdaceb636d2983467d4db3b551bc82d6e031c7.png

with algebra_7045889b6624c02fbceec2d178bd61040f819a00.png, algebra_6df030bb98a7a0593a5d454e1fee20aba8c9d926.png and algebra_b976c44a25fe37b5be6e009bc494a396ec726b28.png.

This decomposition is unique up to reordering of the factors.

Let algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png be an integral domain and let algebra_8e0b24d8a5c6d9a3861e2822654447b4eff6fa8b.png with algebra_2fb79be669b0dd19314e865fa5fac86d032cd33d.png monic.

Then there exists unique algebra_71a1a32b62c29c887d46ca6e74e42b5d3a1dbb29.png such that

algebra_b56c8328878770291f801fd0885bb20be71ec1f7.png

and algebra_4bcc265eb0fd55ea08db7823982cc94cb88f2dc4.png or algebra_cd15c8c326d37542cbee2e3582e2e93ad8e1c9c8.png.

Ideals and Subrings

Let algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png and algebra_391ba25320bcd9d67b1952507e2350803f994ba7.png be rings. A linear map algebra_3e560ecbe48cbd93ee3df1ee5ed9549ddf3139e0.png is a ring homomorphism if the following hold for all algebra_465287e12a60f9244827b01f1b7a4d7602afab6d.png:

algebra_44db9cef1d3bc19a82568c738c51a2933e85fb40.png

A subset algebra_9f3e86478760f2566a928301af15c474f1effba8.png of a ring algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png is an ideal, written algebra_eae3ae7628f48189c8a56807a9caa53dc6e6246e.png, if the following hold:

  1. algebra_6fde3b70a50ab84543ce4818aec81aae37828727.png
  2. algebra_9f3e86478760f2566a928301af15c474f1effba8.png is closed under subtraction
  3. algebra_85f5d497ba6830418e230ff5ec6298999ef5eec3.png for all algebra_f06f19cb0ee81dc2ae2566a4dc103e8e12eaf266.png and algebra_4a6866e3855a59ae16bff6843a7c58d555ddf99e.png, i.e. algebra_9f3e86478760f2566a928301af15c474f1effba8.png closed under multiplication by elements of algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png
    • I.e. we stay in algebra_9f3e86478760f2566a928301af15c474f1effba8.png even when multiplied by elements from outside of algebra_9f3e86478760f2566a928301af15c474f1effba8.png

Ideals are sort of like normal subgroups for rings!

Let algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png be a commutative ring and let algebra_272701e25e6cc93d6cef6397acf8062b8b9d31b5.png.

Then the ideal of algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png generated by algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png is the set

algebra_fbf6a18d1b553c2f5e209f94a5046f1f9bcfe693.png

together with the zero element in the case algebra_8bc07e8447800e943029ced2662f07a8b973a61e.png.

If algebra_b894806d9893a1fa4b0c95b237c166c034b7513e.png, a finite set, we will often write

algebra_766e3b60b434f76a9ff056a7613f06cffe2e7bfa.png

Let algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png be a commutative ring. An ideal algebra_9f3e86478760f2566a928301af15c474f1effba8.png of algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png is called a principal ideal if

algebra_78af10db70afa033a154b763de091978b08f2a13.png

for some algebra_90e50b019feea2055b7588c1cbcf11e7dc496cfc.png.

Let algebra_a9b921fc41c8a6a4288ed5dff486fe71b366e6f4.png be a subset of a ring algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png. Then algebra_a9b921fc41c8a6a4288ed5dff486fe71b366e6f4.png is a subring if and only if

  1. algebra_a9b921fc41c8a6a4288ed5dff486fe71b366e6f4.png has a multiplicative identity
  2. algebra_a9b921fc41c8a6a4288ed5dff486fe71b366e6f4.png is closed under subtraction: algebra_500606dcd4340046c3e46326d9b8d03e0119cdff.png
  3. algebra_a9b921fc41c8a6a4288ed5dff486fe71b366e6f4.png is closed under multiplication

It's important to note that algebra_a9b921fc41c8a6a4288ed5dff486fe71b366e6f4.png and algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png does not necessarily have the same identity element, even though algebra_a9b921fc41c8a6a4288ed5dff486fe71b366e6f4.png is a subring of algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png!

Let algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png be a ring. An element algebra_c2eecb96953f7ff74f50d80ffd389dd697464a24.png is a called a unit if it's invertible in algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png or in other words has a multiplicative inverse in algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png, i.e.

algebra_808da3a39a2233ca6e4179566d39bc3f9abc238f.png

We will use the notation algebra_6ff7ee84f1a658bfad1e3099bc374d0c4cfc1d13.png for the group of units of a ring algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png.

In a ring algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png a non-zero element algebra_cb254ffe3e498884a0e8a2679f6c3a003f7219a5.png is called a zero-divisor or divisor of zero if

algebra_b2bd0d40d16a830e25e5b2a40bd88a4da45d1164.png

An integral domain is a non-zero commutative ring that has no zero-divisors, i.e.

  1. If algebra_67cd7b0e11d3627986f9dff3ebf9759df9a9c912.png then algebra_eebb1783352f1694019ac33ef778643816ce64e6.png or algebra_6004ed990e89192edfd05a4076033c8cbb6ed41f.png
  2. algebra_aab2fdb2687031d3ad22de5eead8ac92d4b01c7c.png and algebra_55cfac9a7b93f7241fc3a00111fa55adf8dc5c71.png then algebra_232d7b2ebef98ff0acc4086c7559117774ca7b39.png

Factor Rings

Let algebra_eae3ae7628f48189c8a56807a9caa53dc6e6246e.png be an ideal in a ring algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png. The set

algebra_3f4d6639ba383eff760a90d6ab324d8c84993c4d.png

is a coset of algebra_9f3e86478760f2566a928301af15c474f1effba8.png in algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png or the coset of algebra_2a61cee9c707231c73a1fcb6992bc4bb552044fe.png wrt. algebra_9f3e86478760f2566a928301af15c474f1effba8.png in algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png.

Let algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png be a ring and algebra_9f3e86478760f2566a928301af15c474f1effba8.png and ideal of algebra_6ee30c172ceb70f3426abc02237d9f23445ae2f9.png.

The mapping

algebra_fafbae860db2c11ada43adcc18bb9f93bb026222.png

Has the following properties:

  1. algebra_d2c5a0be99462ced52b9a85a2d9e110b2e591df9.png is surjective
  2. If algebra_3e560ecbe48cbd93ee3df1ee5ed9549ddf3139e0.png is a ring homomorphism with algebra_f06f35d330b3c8e6fa20285f3fba1c86e47bb268.png so that

    algebra_9923cc34174e9be2e1527bcba99012c251293095.png

    then there is a unique ring homomorphism:

    algebra_c7b03f5e0622ef9404298368d96852ea7ed0b92a.png

Where the second point states that algebra_f017a9b0f9e8a176a3db97e0116f98ff496ac318.png factorizes uniquely through the canonical mapping to the factor whenever the ideal algebra_9f3e86478760f2566a928301af15c474f1effba8.png is sent to zero.

Let algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png and algebra_391ba25320bcd9d67b1952507e2350803f994ba7.png be rings.

Then every ring homomorphism algebra_3e560ecbe48cbd93ee3df1ee5ed9549ddf3139e0.png induces a ring isomorphism

algebra_d7dbe691f514386eaec33ddb5ba75ae3f6be1883.png

Modules

We say algebra_666d2b83616ce27b39869ccca0277bcccc061a44.png is a R-module, algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png being a ring, if

algebra_a0c8b36616da6d220c079ee218a01b3d4a29d75d.png

satisfying

algebra_0e9cc237a806d8ac8bda1dacf9ce5b31a87c5a16.png

Thus, we can view it as a "vector space" over a ring, but because it behaves wildly different from a vector space over a field, we give this space a special name: module.

Important: algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png denotes a module here, NOT manifold as usual.

A unitary module is in the case where we also have algebra_eea9c24a41f017e4c0563bd2d3d24a47e60ffa33.png, i.e. the ring is a unitary ring and contains a multiplicative identity-element.

Let algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png be a ring and let algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png be an R-module. A subset algebra_ecfbe3cef68c328fea5468339248b852573ac336.png of algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png is a submodule if and only if

  1. algebra_4b819d9ccd918778cb2d25f34e89e50c92da0c27.png
  2. algebra_6b16440bd508523ad5ef9494c5f358bcd015b497.png
  3. algebra_8e9c8202596cf36c2dc4dced7d4d25d491a7ea96.png

Let algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png be a ring, let algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png and algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png be R-modules and let algebra_6b91097def918dad01ec80297f10c0ed64d43416.png be an R-homomorphism.

Then algebra_f017a9b0f9e8a176a3db97e0116f98ff496ac318.png is injective if and only if

algebra_76c2c0342f6987ba0724a6dfe470eb684843bf19.png

Let algebra_1de53832fd771a8c65e89b6c211e280ba5689fd6.png. Then algebra_31e41d8f013a9b034ae9684b14d7b85060b824f5.png is the smallest submoduel of algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png that contains algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png.

The intersection of any collection of submodules of algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png is a submodule of algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png.

Let algebra_764f8e36ef329dd7a9365c48ac285daaea5497eb.png and algebra_e2a1e7225fa16c7a1b75c6d2382749aedee6928c.png be submodules of algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png. Then

algebra_1256fb3f1bf4c2cf18de85931226536b4a1d1ffa.png

is a submodule of algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png.

Let algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png be a ring, algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png and R-module and algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png submodule of algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png.

For ever algebra_2fb8ae51e65737676340d2bb2e3abad19bd05418.png the coset of algebra_cb254ffe3e498884a0e8a2679f6c3a003f7219a5.png wrt. algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png in algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png is

algebra_216adcc504bd9509ea01d3a156f0bed765ca4e51.png

It is a coset of algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png in the abelian group algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png and so is an equivalence class for the equivalence relation

algebra_2ead7e1e3ed97cb407db70b8322698598db8644f.png

The factor of algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png by algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png or quotient of algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png by algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png is the set

algebra_ee91fc02b491518993559615a180e61817921b95.png

of all cosests of algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png in algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png.

Equipped with addition and s-multiplication

algebra_b2330f42ce64cb4ffb58c7c91f6b80e6472c0c77.png

for all algebra_4a0816b3a4ccf2a8569db7b4368698e034d43ed7.png and algebra_4a6866e3855a59ae16bff6843a7c58d555ddf99e.png.

The R-module algebra_718abef5e40b7064689935d761aacd08cda96296.png is the factor module of algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png by submodule algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png.

Let algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png be a ring, let algebra_89c1c92593f5a0478673d0ba2e3c9bd22c53effb.png and algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png be R-modules, and algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png a submodule of algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png.

  1. The mapping algebra_2201fd9d026db6197a99ce39994151b5abeaf27f.png defined by

    algebra_83da4111beb2a3c59979cb6825affc5425df7a73.png

    is surjective.

  2. If algebra_8c32363694965eab36e28d807edc6ff3c640dc0b.png is an R-homomorphism with

    algebra_c3a366809eb61cae4179bec4e70110419df3f889.png

    so that algebra_0fcd63b33b5b27932ef94067f50e367c62625024.png, then there is a unique homomorphism algebra_399713ae85e46c434db6e951113710e1fc8058c2.png such that algebra_ea2109f138be0f13eb178f6b31417bf63af77d0a.png.

Let algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png be a ring and let algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png and algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png be R-module. Then every R-homomorphism algebra_6b91097def918dad01ec80297f10c0ed64d43416.png induces an R-isomorphism

algebra_7d4f5735d151b60e57a8907e870a9b93a4faad88.png

Let algebra_6e6f84bca52ff0b2823eac796bb666d60ce0e535.png be submodules of a R-module algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png.

Then algebra_ab9d9977871eda0af750024aad11a3b9b2f6c240.png is a submodule of algebra_42136b5825067adaa491e12fafacdf85f26c319f.png and algebra_3e216af317d8cafdcf4a6474c43a0ec5438d29d6.png is a submodule of algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png.

Also,

algebra_a9d361a6ac3159c41c3d77f54a01ff2c026eb3a1.png

Let algebra_6e6f84bca52ff0b2823eac796bb666d60ce0e535.png be submodules of an R-module algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png, where algebra_0ae18790a7287778670c4bf7370e69d6379250c5.png.

Then algebra_9ba5308692554c92b1db7c7aa99f3b4eb565fa95.png is a submodule of algebra_ad8f24fa4ea37a71d69a649baf5815e88df36903.png.

Also,

algebra_a9e9b039d21e1d159a32b7b55c79049e6beb49a3.png

Determinants and Eigenvalue Reduction

Definitions

An inversion of a permutation algebra_646d5698fb12ffb42d0ef848d1f81003427a5e97.png is a pair algebra_78e298b10b36bc807f89a3694bb5f50818e2d188.png such that algebra_0b419e220cc4512e7e6958505e78b788f46d068f.png and algebra_2869da3156cfc0fcab526e9f10395e2345c3779b.png.

The number of inversions of the permutation algebra_8a303c22b5ff8640fb384bd5649f789459e04e8b.png is called the length of algebra_8a303c22b5ff8640fb384bd5649f789459e04e8b.png and writtein algebra_27a738a68c39b786c56400fc8cc2d77dce327530.png.

The sign of a permutation algebra_8a303c22b5ff8640fb384bd5649f789459e04e8b.png is defined to be the parity of the number of inversions of algebra_8a303c22b5ff8640fb384bd5649f789459e04e8b.png:

algebra_c4f36924e24389e731ad83c434f442f4c1d9aee5.png

where algebra_81f72bec7ef4bf9d1e66ff4ef8af562fb25c5473.png is the length of the permutation.

For algebra_b1f271c6d4554ce07464f29a42f953e4a65d053c.png, the set of even permutations in algebra_cd684f0319333005c3661765ad2d76359ab41ff2.png forms a subgroup of algebra_cd684f0319333005c3661765ad2d76359ab41ff2.png because it is the kernel of the group homomorphism algebra_d9b97c92de2bb08ab8dcf7f672b22495c950e9eb.png.

This group is the alternating group and is denoted algebra_7b0f754764e5176e8b327f85797b19e418b26be1.png.

Let algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png be a ring.

The determinant is a mapping algebra_b8932f506ab638627945546e2da2dbd77fe88fe8.png given by

algebra_90af892584fbe444c3ca24644310bf99e06825a6.png

Let algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png be a commutative ring, then

algebra_be82fa20f15b389355bcdb2ed75d10b9f5eeaf11.png

Let algebra_3cf647f5eca479fa8a18df5b958f095a7f4f45df.png be an algebra_fda03f959d8b165fea0b26a0869d140ab6cb0f2c.png matrix with entries from a commutative ring algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png.

For a fixed algebra_b7c30b8be6c76b11192e84f50168a1541d5fab2d.png the i-th row expansion of the determinant is

algebra_cfe355a44bb23a8a4bf2f6e66fe1adc3155a3434.png

and for a fixed algebra_79974a5eda6ec26cfbba2e72f6c8f39ada1ab9f5.png the j-th column expansion of the determinant is

algebra_fbaac6b5b7e4fbd0ed479aa4f460341bc6c305e3.png

where algebra_79faf80e2458d3aab814683ec09bf65de95ee85f.png is the algebra_78e298b10b36bc807f89a3694bb5f50818e2d188.png cofactor of algebra_f75be59687ef2da76b04e88af02585ab68efc56c.png

algebra_7824391d78126fedb75e0fed38b93e6e889aadba.png

where algebra_de8bdc21c28ae283e0d2f25f6e6269e591bda1f0.png is the matrix obtained from deleting the i-th row and j-th column.

Cayley-Hamilton Theorem

Let algebra_2e05b06949d0d78c3cb188c2225901372fcc9593.png be a square matrix with entries in a commutative ring algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png.

Then evaluating its characteristic polynomial algebra_798c5d5e635a51bbd428c4c4269fc7ebfceb3c57.png at the matrix algebra_f75be59687ef2da76b04e88af02585ab68efc56c.png gives zero.

Eigenvalues and Eigenvectors

Theorems

Each endomorphism of non-zero finite dimensional vector space over an algebraically closed field has an eigenvalue.

Inner Product Spaces

Definitions

Inner product

An inner product is a (anti-)bilinear map algebra_e88501fb9eeefc725e213499c1a810dc0710e157.png which is

  1. Symmetric
  2. Non-degenerate
  3. Positive-definite
Forms

Let algebra_41c048429523113c3c06eb8a007a50f609f04f55.png. We say algebra_86504a08fe587f7765c3c6e108c08ed5edee64f8.png is a bilinear form if

algebra_5906bef24dfd5ca405a09af977be80e2db21a558.png

for algebra_5ad51b2820c6180eb76f0cbc77db06185f6ef8bf.png, algebra_3a619d5a5cfd323dd0a1d30296ca76670254125c.png and algebra_dc0f9c7ae2242455e20af4cb8cde4bfecb305f4d.png.

A symmetric bilinear form on algebra_138fc0328ebd60fa7cec6af387a8682f611cdc54.png is a bilinear map algebra_38735932ff7d7538b29660338307fd374919f9fa.png such that algebra_d28121173305f9c01a9cc996722054f9eda1fd17.png.

Given two 1-forms algebra_e545c466c15f527ef6e921ba62a0ae8b78a574c1.png at algebra_d78e7eeb9c43315055c5372165a3d1f7c894e7af.png define a symmtric bilinear form algebra_02eb354b18ee7cb3f7ca04af8c42150fdcabdf72.png on algebra_39b60f854ea2dc6a2f51faebd5d0d42c36f1bca5.png by

algebra_c52056219a3f0de7c6d578c4dd9e99e41d94a9f0.png

where algebra_a199b245b4562cc0654b60e6041079c806a78774.png.

Note that algebra_fca133a4e128274caef56d4d6a1ef4b011d6e046.png and we denote algebra_cce09f46b4914f1927278fbb0876dfa169f11bac.png.

REDEFINE WITHOUT THE ADDED TANGENT-SPACE STRUCTURE, BUT ONLY USING VECTOR SPACE STRUCTURE.

A symmetric tensor on algebra_ec1d6ffeea14c7290c9b899268d2f1e6598a027c.png is a map which assigns to each algebra_d78e7eeb9c43315055c5372165a3d1f7c894e7af.png a symmetric bilinear form on algebra_138fc0328ebd60fa7cec6af387a8682f611cdc54.png; it can be written as

algebra_a0a33bebddd528c9760a4e1df4dd2f877900a285.png

where algebra_1c76c602954c06be3a41a3e254b12153b0d872d3.png are smooth functions on algebra_ec1d6ffeea14c7290c9b899268d2f1e6598a027c.png.

Remember, we're using Einstein notation.

Skew-linear and sesquilinear form

We say the mapping algebra_b314324651ef278db247a6282c2c7967d1e6c28d.png between complex vector spaces is skew-linear if

algebra_938bc5e2993b9f43b188cd07e91c3d358ffe3c83.png

Let algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png be a vector space over algebra_552ebfb02492c81923c04dc8e94bf78e5c966661.png equipped with the inner product

algebra_9d4120067b8f91d78316e8ff24819dbe74404fd0.png

Since this mapping is skew-linear in the second argument, i.e.

algebra_2f14b6454d10c8f15162465fda344c5a27140625.png

we say this is a sesquilinear form.

Hermitian

Let algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png be a vector space over algebra_552ebfb02492c81923c04dc8e94bf78e5c966661.png. If the sequilinear form algebra_cc7d0bc808720aec3b7cefd12819fb86dc0362e8.png is symmetric, i.e.

algebra_4f757d6441beacecac21dbd838a6736ad6fffbee.png

we say algebra_0771ce7b70bacdbe91b857a46becb2dd6d7e0e92.png is a Hermitian.

Theorems

Let algebra_894a7bf262bc8b225e430f3960c5fd75cff711d9.png be a vectors in an inner product space. Then

algebra_af656e9b5ef81e7c514097d744483cfc136331cf.png

with equailty if and only if algebra_a30062e7b2cbc08eebe63686f520106acab766d4.png and algebra_4826159b7fea17a9828a89db6af3e11af01aa08c.png are linearly dependent.

Adjoints and Self-adjoints

Let algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png be an inner product space. then two endomorphism algebra_228f6f16943a7a14a257644403d74d68567ddedb.png are called adjoint to if the following holds:

algebra_28f7199c29138e44e9ce1ea948384bb89ffe94e8.png

Let algebra_62b44a9e51a9310bb6c753993596ba82a99fe1c5.png and algebra_ddfe50844fc29a17bc64da6a92eee4b9582cfe18.png be the adjoint of algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png. We say algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png is self-adjoint if and only if

algebra_775b75ab3c3c7f16adf26a4e55041e0a079b82bd.png

Let algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png be a finite-dimensional inner product space and let algebra_62b44a9e51a9310bb6c753993596ba82a99fe1c5.png be a self-adjoint linear mapping.

Then algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png has an orthonormal basis consisting of eigenvectors of algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png.

We'll prove this using induction on algebra_40c030a30a772542ae8bbe40874513029dfb1d5f.png. Let algebra_437d5f28745b4ccc930c4342eabaa6d220416ea9.png denote the an inner product space with algebra_c95c637537ce7180df94ad927ff9bdafe3b71003.png with the self-adjoint linear mapping algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png.

Base case: algebra_4b3efbce269a53f6a87631151ae7808410b24874.png. Let algebra_7300b4d634dfea3f0f85310da5986fa913ee1cec.png denote an eigenvector of algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png, i.e.

algebra_fed85ccb6fcb9371e2a87afe3fe77188b67b5ae8.png

where algebra_d5c60a4ae2b31ef1ab0bdd9fc92339f09358331b.png is the underlying field of the vector space algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png. Then clearly algebra_e852cdb6d6c19d2425df388ccc13bc978c76f41e.png spans algebra_f377695d881c4f672d1bf48b016adc1c0890005c.png.

General case: Assume the hypothesis holds for algebra_c6a86f135f24914f9f9aab38c2f7c87c5b426ea7.png, then

algebra_118d667215086b1b0ee11e2170623623fb67e56d.png

where

algebra_7461b0bc933f317e6307249811c43f1b70eac977.png

Further, let algebra_1c4128044160304759acbc6074b6964d678c7f9d.png and algebra_2d10bde726a4e60a9ccdd78736fafb2c541c2f2f.png with algebra_ff6c78ac6e983cc2431fa84310d0b89f9a01f7e0.png be an eigenvector of algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png. Then

algebra_7c32111a80f96c39ea0ce30a89c1e9bd21179982.png

since algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png is self-adjoint. This implies that

algebra_c339d574b9ff890cb2580af4bc3037bcfb808d68.png

Therefore, restricting algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png to algebra_88588441d42377b6f22381e30ad3663de48b1d0e.png we have the mapping

algebra_8fc3e6b302b5ddade2f870fa89971115d80d2d6b.png

Where algebra_bee236ab86596a888585414ca3a8b64226906160.png is also self-adjoint, since algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png is self-adjoint on the entirety of algebra_3d541add5ce170d63d8fff13b4ab812e4aba7d23.png. Thus, by assumption, we know that there exists a basis of algebra_88588441d42377b6f22381e30ad3663de48b1d0e.png consisting of eigenvectors of algebra_bee236ab86596a888585414ca3a8b64226906160.png, which are therefore orthogonal to the eigenvectors of algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png in algebra_437d5f28745b4ccc930c4342eabaa6d220416ea9.png. Hence, existence of basis of eigenvectors for algebra_437d5f28745b4ccc930c4342eabaa6d220416ea9.png implies existence of basis of eigenvectors of algebra_3d541add5ce170d63d8fff13b4ab812e4aba7d23.png.

Thus, by the induction hypothesis, if algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png is a self-adjoint operator on the inner product space algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png with algebra_e5fcb072a5aa82d2e2f128cd71132542bf6e9afe.png, algebra_b1f271c6d4554ce07464f29a42f953e4a65d053c.png, then there exists a basis of eigenvectors of algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png for algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png, as claimed.

Jordan Normal Form

Notation

  • algebra_71fa22f29b8955ca4446af403e21032062d84c03.png is an endomorphism of the finite dimensional F-vector space algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png.
  • Characteristic equation of algebra_1827136eeff990e82cb2b9950aa049a32cc2f910.png:

    algebra_f308b0e4debf4ba3f7ee12693975d1a0f009bbb2.png

  • algebra_54ce5b9644b02f2b85e137ea7a677b4a4df30714.png
  • Polynomials

    algebra_6d487f547f069428f1f84fcdedbe4598b4d10659.png

  • Generalized eigenspace of algebra_1827136eeff990e82cb2b9950aa049a32cc2f910.png wrt. eigenvalue algebra_2115ea8fa5eac155ca4ddf281b58f34ee40f7fd8.png

    algebra_30c5a09f1bf4ca4dcd429f6a1f3c5f1acb0785e2.png

  • algebra_eb0cac4d9ceb2c8c0a6d6c6aabe2a2bf03570ee3.png denotes the dimension of algebra_5475e541ffaaee18e0d2eab4dd79260000959396.png
  • Basis

    algebra_43bcf0182b9ef814b68ae1e171445752e40ce1c0.png

  • Restriction of algebra_1827136eeff990e82cb2b9950aa049a32cc2f910.png

    algebra_9507d1dd9deafd8ef532a00ca9679968700dbf33.png

  • algebra_25fac79b8e84d0e03c8d4057821006d84076b454.png defined

    algebra_d602995ecbb4e2ae2c35ff382c1711670049a76b.png

    is well-defined and injective.

  • algebra_c266fd4ef45d28a7d621ce8a32c4594832e06d3a.png

Definitions

An endomorphism algebra_97409c1d3d662671f832564ad597c4d4db5d0127.png of an F-vector space is called nilpotent if and only if algebra_ffaa209caf1fd61764980c0d962b6a47072a44ff.png such that

algebra_7c1a4368e4bbe7144a48d3283a9b1aa2b79e906e.png

algebra_3ec937c79f7bddc072142f0c1332a0499a9ee6ce.png

Motivation

  • algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png be a finite dimensional vector space
  • algebra_97409c1d3d662671f832564ad597c4d4db5d0127.png an endomorphism
    • a choice of ordered basis algebra_49af2ec32d071a93d42ea9f6cba4ef37624c1e14.png for algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png determines a matrix algebra_28bcf7ce0a2c6711e5d6144286ec02fd3c1fbd0c.png representing algebra_f017a9b0f9e8a176a3db97e0116f98ff496ac318.png wrt. basis algebra_49af2ec32d071a93d42ea9f6cba4ef37624c1e14.png
  • Another choice of basis leads to a different representation; would like to find the simplest possible matrix that is conjugate to a given matrix

Theorem

Given an integer algebra_30ee5f05ecee8a8b0278aa51bbaf9e75b443abd2.png, we define the algebra_e76ddd2a88dcd5232f659db5da4598e2d69c3732.png matrix algebra_65a06ac298a043347d491250adb3201c12c21fd6.png

algebra_9d344c8a47b292e6159bf7fb29f574db4594a1b6.png

or equivalently,

algebra_f26d1579e601be58254d92ba7b4b2b13743daed1.png

which we call the nilpotent Jordan block of size algebra_6ee30c172ceb70f3426abc02237d9f23445ae2f9.png.

Given an integer algebra_30ee5f05ecee8a8b0278aa51bbaf9e75b443abd2.png and a scalar algebra_ba6f2a9ca9c40226ba21ccd7ec9866712fb9d982.png define an algebra_e76ddd2a88dcd5232f659db5da4598e2d69c3732.png matrix algebra_462846f70396e371153fac394dcecab0d9975d54.png as

algebra_2d9fbdcff381849577247ae55536ea65f01c530c.png

which we is called the Jordan block of size algebra_6ee30c172ceb70f3426abc02237d9f23445ae2f9.png and eigenvalue algebra_d2a98a288f5c0c00f2bb7659dbf786e35d944225.png.

Let algebra_d5c60a4ae2b31ef1ab0bdd9fc92339f09358331b.png be an algebraically closed field, and algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png be a finite dimensional vector space and algebra_71fa22f29b8955ca4446af403e21032062d84c03.png be an endomorphism of algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png with characteristic polynomial

algebra_4b318883e7be389eca3da8704cc4de620a2e44eb.png

where algebra_c2bb1d53eee78655294cef52ae01ab6b203f5865.png and algebra_009ad1e45fd904e7e2b3a163bb24b659d4058db3.png, for distinct algebra_3a1d5241d3fa4421472f9d73aec77a3a3d636d7b.png.

Then there exists an ordered basis algebra_49af2ec32d071a93d42ea9f6cba4ef37624c1e14.png of algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png s.t.

algebra_e6b534bd7ff48f4416df941c31448e2cfddbd216.png

with algebra_fd9e0c583d252cf4a06013fb337a16d38659d6dd.png such that

algebra_fc2c3de4cafa34c9bc923b03b576f1e292564926.png

with algebra_befddd7b46d4e8afd763aaddb523b800fc5c639b.png.

That is, algebra_1827136eeff990e82cb2b9950aa049a32cc2f910.png in the basis algebra_49af2ec32d071a93d42ea9f6cba4ef37624c1e14.png is block diagonal with Jordan blocks on the diagonal!

Proof of Jordan Normal Form
  • Outline

    We will prove the Jordan Normal form in three main steps:

    1. Decompose the vector space algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png into a direct sum

      algebra_d7b1ff22fa94f77d37e1624a2fa4541660b16e25.png

      according to the factorization of the characteristic polynomial as a product of linear factors:

      algebra_23c6d70afea374b1dee9f89975e8139a419a4690.png

      for distinct scalars algebra_3a1d5241d3fa4421472f9d73aec77a3a3d636d7b.png, where for each algebra_b7c30b8be6c76b11192e84f50168a1541d5fab2d.png:

      • algebra_54ce5b9644b02f2b85e137ea7a677b4a4df30714.png
      • algebra_1c13c8e74212e1e05578aa9e28149ea7088b3f75.png
    2. Focus attention on each of the algebra_e4c78347a4f449021300d4e53543634109142892.png to obtain the nilpotent Jordan blocks.
    3. Combine Step 2 and 3
  • Step 1: Decompose

    Rewriting algebra_cd4dc6b947cd605c44597dd151b4604d84ed1a86.png as

    algebra_182e2ea45e1db5690f263affe84e7a8a105e187f.png

    where algebra_1efa1df506cbcb2db0b273389778320c8fd8de1f.png are the eigenvalues of algebra_1827136eeff990e82cb2b9950aa049a32cc2f910.png.

    For algebra_befddd7b46d4e8afd763aaddb523b800fc5c639b.png define

    algebra_6d487f547f069428f1f84fcdedbe4598b4d10659.png

    There exists polynomials algebra_3913fba41162f31d53495fd9828eda91da9465ba.png such that

    algebra_425a1889681f7eb6207fab2cbfa8c5c54fbd7325.png

    For each algebra_befddd7b46d4e8afd763aaddb523b800fc5c639b.png, let

    algebra_537e3a65c7012a8a900af2053c3eae2b768cf8e9.png

    be a basis of algebra_5475e541ffaaee18e0d2eab4dd79260000959396.png, where algebra_eb0cac4d9ceb2c8c0a6d6c6aabe2a2bf03570ee3.png is the algebraic multiplicity of algebra_1827136eeff990e82cb2b9950aa049a32cc2f910.png with eigenvalue algebra_2115ea8fa5eac155ca4ddf281b58f34ee40f7fd8.png.

    1. Each algebra_5475e541ffaaee18e0d2eab4dd79260000959396.png is stable under algebra_1827136eeff990e82cb2b9950aa049a32cc2f910.png, i.e algebra_a5d1d2c1dd757874c41bf80961812291b5457dc7.png
    2. For each algebra_964f5c0895be0566e8f3bc7cc6084dab9ed9164f.png, algebra_e84a0c14fe55dbc8d9d647c27e1adeb5264fc8b3.png such that

      algebra_4a1337cf6d2c3609f7dec9f35a394a25f29044de.png

      In other words, there is a direct sum decomposition

      algebra_750bca146c0409b981dd0f2823cc0bea7486221e.png

    3. Then

      algebra_6161f21c9aed59608a2e72ec6c8b112b5aa4c9a7.png

      is a basis of algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png, so in particular algebra_5eb6c5e6fa25fb8bd9dcc48bd62c0d91d50d573a.png. The matrix of the endomorphism algebra_1827136eeff990e82cb2b9950aa049a32cc2f910.png wrt. to basis algebra_49af2ec32d071a93d42ea9f6cba4ef37624c1e14.png is given by the block diagonal matrix

      algebra_2518866d8fc5b50599fc8d1fd922a2a8c3c2a8f4.png

      with algebra_d8909b695b98fa1bc2b9705766b262c14b125129.png.

    1. Let algebra_1f50eedb98039437138e5d16684c2c75035e5258.png is that

      algebra_32c11c443cdef036f8c9a11b667c461b31ddd58f.png

      Then

      algebra_7f164fa9a79033121e7b2268e8a460a3d536ff75.png

      Hence algebra_81833155d91283de17f86eba78e88f252d8600c1.png, i.e. algebra_5475e541ffaaee18e0d2eab4dd79260000959396.png is stable under algebra_1827136eeff990e82cb2b9950aa049a32cc2f910.png.

    2. By Lemma 6.3.1 we have algebra_15b3cd8f3e30922042c277f89d60fe816e462754.png and so evaluating this at algebra_1827136eeff990e82cb2b9950aa049a32cc2f910.png, we get

      algebra_f6641af85b691f147650e2236d35fc18df6d62ba.png

      Therefore, algebra_f232d891910a0d35427f55a1c1520f131d3c896e.png, we have

      algebra_d6bcaba1d2ad63a56edb3f55a0508b6ce668d7a2.png

      Observe that

      algebra_5fdba1e6bbc293570fe880acd54dbff18d48c03c.png

      where we've used Cailey-Hamilton Theorem for the second equality. Let

      algebra_762dc182d3e33010141c63a3b5a547cdfffba5bf.png

      then

      algebra_d3d5243110582fb9b580126edd7dc1f8daafba5f.png

      hence all algebra_964f5c0895be0566e8f3bc7cc6084dab9ed9164f.png can be written as a sum of algebra_db8abd982750fbc639515f463392e1ebfbc56d43.png, or equivalently,

      algebra_5c42043e842d0ca76d02453c76dbcbd84be4522e.png

      as claimed.

    3. Since algebra_1f79e8cf189b67bc8aad59f13ac17f07a0fd55e0.png is a basis of algebra_5475e541ffaaee18e0d2eab4dd79260000959396.png for each algebra_b7c30b8be6c76b11192e84f50168a1541d5fab2d.png, and since

      algebra_60254a516d15f78579edec709f1ae4caba3fd462.png

      we have

      algebra_3958ee6b1d6b8c1c6204cda17ef6da7763b7fd98.png

      form a basis of algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png. Consider the ordered basis algebra_23ef323437416935f2a48afb9b45f4f762fcaa6e.png, then the algebra_fab5eb132ce512fd6bbd47110ae012739381c5e0.png can be expressed as a linear combination of the vectors algebra_d2bb9b6b694a8fd9d8360893ca22cfc62f8202b4.png with algebra_ca1b1b296e0e9e34c43a5e86a0fd14a2ddcd03b3.png. Therefore the matrix is block diagonal with i-th block having size algebra_8cbb5fd005d7337d5f70137717a731537564a103.png.

    From this, one can prove that any matrix algebra_f75be59687ef2da76b04e88af02585ab68efc56c.png can be written as a Jordan decomposition:

    Let algebra_3e09190e36d84bcc098fd32589b5f4f9861c000e.png, then there exists a diagonalisable (NOT diagonal) matrix algebra_ec1d6ffeea14c7290c9b899268d2f1e6598a027c.png and nilpotent matrix algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png such that

    algebra_863400a8ab8ada918add6bf81fdb2ad8e554ef6e.png

    In fact, this decomposition is unique and is called the Jordan decomposition.

  • Step 2: Nilpotent endomorphisms

    Let algebra_3bd1f26e899259191c15988046ebcac426ab307c.png be a finite dimensional vector space and algebra_9382844420b8420780cd451c223d21db2e8df36a.png such that

    algebra_793c06e0f4d43fbe2670bd5028cfbe8c374566d7.png

    for some algebra_03d562610fbb53ec50fe1224fa55faf0d6c29492.png, i.e. algebra_06533eba579f5050f2511b94841d9efe169c005f.png is nilpotent. Further, let algebra_8f44de754519a6b4b737f2e24c2083a1a7cb4a03.png be minimal, i.e. algebra_1e8fcf83a973fdb9564eba40492ba2d86706be09.png but algebra_512dd7de880d177c623222ed0bf56569d553a61c.png.

    For algebra_bbdd91105300e7a8612cf5e071dcbc25f36e4e67.png define

    algebra_b8e260c768e22f1bbb2be60cd542149ad92c6e62.png

    If algebra_e603a43b9ae6def6ea96f35bcd208e4f4339fdbd.png then

    algebra_791473e7249259047a2ae017f3c727c10c3c02b8.png

    i.e. algebra_cd92feb6bb19ef2bd20acbdff067e476b4f6a7f6.png, hence

    algebra_2b7fc2f4e68dc88a670b030a07b6bce687654a05.png

    Moreover, since algebra_57310cb3b174edd433e64c0b7cc0ff1bad06eb98.png and algebra_45fe7a8f00bea192551b007a3c46c13ebda0d1e5.png, we have algebra_4164549414d837ab9722229e3d626012b8fa2c0f.png and algebra_a65a5fc98894e15ac45d1af442307b0de4e36d72.png. Therefore we get the chain of subspaces

    algebra_25114c7e822e74ea5c52a9d603545841e3d47095.png

    We can now develop an algorithm for constructing a basis!

    • Constructing a basis
      1. Choose arbitrary basis algebra_5d16132ce66d26dbe435de4eb37789f42402ef30.png for algebra_07d3a80132a17547c867bc83cb47d4354e17a91e.png
      2. Choose basis of algebra_663bb0e9baf0b5f5c3978ae31ec52fca11fc26c6.png of algebra_817628f213b203517e549380b6935e88875926ea.png by mapping algebra_5d16132ce66d26dbe435de4eb37789f42402ef30.png using algebra_ec06ce567873874fbe4e681c48164a3c67088a4c.png and choosing vectors linearly independent of algebra_f7a592f08104be13b42090a0e510df266432bda0.png.
      3. Repeat!

      Or more accurately:

      1. Choose arbitrary basis for algebra_07d3a80132a17547c867bc83cb47d4354e17a91e.png:

        algebra_387e701ae6d5b4734a4703d132d664b793310312.png

      2. Since algebra_0edbf68f1f255f655c75b012c0b171e5ed327be9.png is injective, by the fact that the image of a set of linear independent vectors is a linearly independent set if the map is injective, then

        algebra_900d57180e7cbdb57f86a6483c1c2895f43a6376.png

        is linearly independent.

      3. Choose vectors

        algebra_b70897dfe8df58a35be07845eb31b67ca9f20536.png

        such that

        algebra_baca674b08a13dc5a4f6e237ba5e50ae2a951db2.png

        is a basis of algebra_d9d9a782d10e4702d26d5697a362e6cc3e41b699.png.

      4. Repeat!

      Now, the interesting part is this:

      Let algebra_3bd1f26e899259191c15988046ebcac426ab307c.png be a finite dimensional vector space and algebra_9382844420b8420780cd451c223d21db2e8df36a.png such that

      algebra_793c06e0f4d43fbe2670bd5028cfbe8c374566d7.png

      for some algebra_03d562610fbb53ec50fe1224fa55faf0d6c29492.png, i.e. algebra_06533eba579f5050f2511b94841d9efe169c005f.png is nilpotent.

      Let algebra_49af2ec32d071a93d42ea9f6cba4ef37624c1e14.png be the ordered basis of algebra_3bd1f26e899259191c15988046ebcac426ab307c.png constructed as above

      algebra_b4261aa7281fe4de5e2e1789bc78241cf5e55ba5.png

      Then

      algebra_e6a55ed2ae2da40678aa7e638d20c85d8a88661c.png

      where algebra_65a06ac298a043347d491250adb3201c12c21fd6.png denotes the nilpotent Jordan block of size algebra_6ee30c172ceb70f3426abc02237d9f23445ae2f9.png.

      It follows from the explicit construction of the basis algebra_49af2ec32d071a93d42ea9f6cba4ef37624c1e14.png that

      algebra_3f18637209a312d7ded24021cd026e0cabbece8f.png

      Since algebra_a19a419201cb51ab26bdc838efb1acd36269648c.png is defined by how it maps the basis vectors in algebra_49af2ec32d071a93d42ea9f6cba4ef37624c1e14.png, and in the basis algebra_49af2ec32d071a93d42ea9f6cba4ef37624c1e14.png algebra_3a6d9a11e48fab76e67027d85c3ea0daac75c9c5.png becomes a vector of all zeros except the entry corresponding to the j-th basis-vector chosen for algebra_18c84a6622c9a88fda7b41ad1ffd9d0fe9441018.png, where it is a 1.

      Hence algebra_a19a419201cb51ab26bdc838efb1acd36269648c.png is a nilpotent Jordan block as claimed.

      Concluding step 2; for all nilpotent endomorphisms there exists a basis such that the representing matrix can be written as a block diagonal matrix with nilpotent Jordan blocks along the diagonal.

  • Step 3: Bringing it together

    Again considering the endomorphisms algebra_a22b4fbe4886f6e8ae9468b629a9de1b608bc3d8.png restricted to algebra_5475e541ffaaee18e0d2eab4dd79260000959396.png, we can apply Proposition 6.3.9 to see that this endomorphism can be written as a block diagon matrix of the form stated for a suitable choice of basis.

    The endomorphism algebra_1eabc3c4880a4b846a87dc19bc637e31c18b40de.png restricted to algebra_5475e541ffaaee18e0d2eab4dd79260000959396.png is of course algebra_1565ef60008dbde003c1616515bf6989465bde30.png, thus the matrix wrt. the chosen basis is just algebra_623fb758f922c203c4e5f45467f1ced7b4dcf23b.png. Therefore the matrix for algebra_05a24dd118fa5ddfa651f4de4ba7cbc3def77a0b.png (when restricted to algebra_5475e541ffaaee18e0d2eab4dd79260000959396.png) is just algebra_623fb758f922c203c4e5f45467f1ced7b4dcf23b.png plus algebra_7784f0cc1859aa7f2560e44d80fe8a0b2501cd21.png, i.e. algebra_950b8686c9806bb744bb244307808b37e563ed0f.png.

    Thus, each algebra_3ea5aa820c8e881928629f7f2762e2620ddf0bb7.png appearing in this theorem is exactly of the form we stated in Jordan Normal form.

Algorithm

  1. Calculate the eigenvalues, algebra_b6ec9d838362653ed2d8f4a1414883340b84dd34.png of algebra_f75be59687ef2da76b04e88af02585ab68efc56c.png

    algebra_2bd1f87c1a537648bb8ae02d821b77db5a1cee93.png

  2. For each algebra_b7759700b0b146320febf8598a5882528c71620d.png and algebra_953291eb01b4ebda6895eca4c93029f87517d5f8.png, let

    algebra_906f68a09a9f1129a485057b212e5fa0269d5901.png

    1. Compute algebra_bd9929e0018e8ffd5e0cf1e94183a0c44df51127.png and

      algebra_5e5f5053cb3172bfb67726dcaa12af3a66a4562f.png

    2. Let

      algebra_daa8a2ce13fe6c1dacc553fd9f8e37b348101131.png

    3. Set

      algebra_9097d9886301448c5618c5dc1c913b06895c175c.png

      i.e. the difference in dimension between each of the nullspaces.

      1. Let algebra_79974a5eda6ec26cfbba2e72f6c8f39ada1ab9f5.png be the largest integer such that algebra_ede136f24f195865fb2c9f74b33d32469f726557.png.
      2. If algebra_c2874faf7fe3cd2ca14c29973923fa1d29e24745.png does not exist, stop. Otherwise, goto step 3.
      3. Let algebra_647f3ad8f4b71a946f9d1501a1e3d10a97bcc4b9.png and algebra_d88587f2d852cf4426ec8e461e4c82985b9ba349.png.
      4. Let

      algebra_ee75520c88867ebd00bdc757b8e22f27ce6108cd.png

      1. Change algebra_0900d694997fefcda2fa6f43235a1da9b62c4f76.png to algebra_cac4e0b0eebc77876e3bfbdd40213343b4548011.png for algebra_3f308108e47f6039788816b0d281af6a34fdc948.png.
  3. Let the full basis be the union of all the algebra_49af2ec32d071a93d42ea9f6cba4ef37624c1e14.png, i.e. algebra_a60440deee34bba47dcae692c648ad03f400ed2d.png.

Tensor spaces

The tensor product algebra_5e8002a6a49554c56f2d6061bf740617420e3787.png between two vector spaces algebra_07eb6a6d51bab611982313d706c4993ed7dd9d45.png is a vector space with the properties:

  1. if algebra_d3a5f477ab561e6c28f8c617cc3289075d9928e1.png and algebra_2c471691859541d6ae52b343eab670cb756a9844.png, there is a "product" algebra_2460dccae5f46928780fd4c1fc1204c7ce4d8f60.png
  2. This product algebra_02664a30d82346ecc98ca0bf04c01e8a17c983d8.png is bilinear:

    algebra_8e7df6d16c6c99c9911a541549bb99f4f374818f.png

    for algebra_5ad51b2820c6180eb76f0cbc77db06185f6ef8bf.png, algebra_3a619d5a5cfd323dd0a1d30296ca76670254125c.png and algebra_dc0f9c7ae2242455e20af4cb8cde4bfecb305f4d.png.

Let algebra_f8a2b400bf9def7b5b67035dfb5f06e148ab7065.png be a vector space.

If algebra_6974e16e1f5c1ca213551154847fa044dde0d5ea.png is a bilinear map, then there exists a unique linear map algebra_fe050b8b3c922ff51cf5d7a976dc82bbc056c175.png such that

algebra_b1b8fba4fe05ebf959e06b3bb029f0c8be34df09.png

I find the following instructive to consider.

In the case of a Cartesian product, the vector spaces are still "independent" in the sense that any element can be expressed in the basis

algebra_693e4f2d85eb1851dd86ca5676e722fabd8c2f03.png

which means

algebra_3b8fa7a99c88091176ab97a8e81db9c52eb20bd8.png

Now, in the case of a tensor product, we in some sense "intertwine" the spaces, making it so that we cannot express elements as "one part from algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png and one part from algebra_3bd1f26e899259191c15988046ebcac426ab307c.png". And so we need the basis

algebra_52e4a01fa312af3fb0af426dcfdb4f14054b9303.png

Therefore

algebra_abaf5159230b4908977bb5d2d4128d1b6194bdd8.png

By universal property of tensor products we can instead define the tensor product between two vector spaces algebra_07eb6a6d51bab611982313d706c4993ed7dd9d45.png as the dual vector space of bilinear forms on algebra_22cf71de0db1a8b88e9290675d9aa8ef8c042a3a.png.

If algebra_d3a5f477ab561e6c28f8c617cc3289075d9928e1.png, algebra_2c471691859541d6ae52b343eab670cb756a9844.png, then algebra_2460dccae5f46928780fd4c1fc1204c7ce4d8f60.png is defined as the map

algebra_474c3feb348ecfa6f6bfe5b45bee43bfe4ee381c.png

for every bilinear form algebra_86504a08fe587f7765c3c6e108c08ed5edee64f8.png.

That is,

algebra_91aea40e0aba3fb5d359980edc7e8c9046762226.png

and

algebra_b3711a2786ce63bf2e1dc40961acab6623456d58.png

by

algebra_3fc64af87f73ac213c0a1f232b16e68eda86ca30.png

Observe that this satisfies the universal property of tensor product since if we are given some algebra_6974e16e1f5c1ca213551154847fa044dde0d5ea.png, for every algebra_abe73d5b263a67fe9c7f7434e9d60268fbb26890.png, i.e. algebra_0da2ee3b52f2b774406273416ed710bd81df7f36.png, we have algebra_acb9876241cb272453f33a93334b9391ed7b65a4.png, i.e. algebra_494c88acbf10a0ae55301b726a8dff0e60e41a49.png is a bilinear form on algebra_22cf71de0db1a8b88e9290675d9aa8ef8c042a3a.png. Furthermore, this dual of bilinear forms on algebra_22cf71de0db1a8b88e9290675d9aa8ef8c042a3a.png is then algebra_64c531fa234b20c6804b0f1ed797fa0ec5ba5722.png (since we are working with finite-dimensional vector spaces).

algebra_13a1a2398ba7e4bd6f06faa8f2f285fd9f4d5a74.png

From universal property of tensor product, for any algebra_6974e16e1f5c1ca213551154847fa044dde0d5ea.png, there exists a unique algebra_fe050b8b3c922ff51cf5d7a976dc82bbc056c175.png such that

algebra_5e7111c0b521517e540b5ee8c625c66b9348c0cc.png

Letting the map

algebra_6281456950b783dc9fa060087ab0b478575ee555.png

be defined by

algebra_b7d73912747ff7e2cbc375b0eabcac85e00dc335.png

where algebra_bccac2b8c5607d5325a7c1267368c3bcfc82ea5c.png is defined by algebra_019175af92ceb52a735efb5a6f83ccf42c4a76cf.png. By the uniqueness of algebra_31aa0b82f4e4fe4be2c026d5eddd655a517a7f53.png, and linearity of the maps under consideration, this defines an isomorphism between the spaces.

Suppose

  • algebra_576867d6fb3c9a5b78e8e1851c251f57b1340ac9.png is a basis for algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png
  • algebra_5eb3b46c64aac82dd98e718048f24d56b35acfc7.png is a basis for algebra_3bd1f26e899259191c15988046ebcac426ab307c.png

Then a bilinear form algebra_41c048429523113c3c06eb8a007a50f609f04f55.png is fully defined by algebra_6952b3082b182c7773835ffb47be373d7057db7a.png (upper-indices since these are the coefficients of the co-vectors). Therefore,

algebra_9c8aa87bdaa19b5cf9a811f4aff14935b8ca224b.png

Since we are in finite dimensional vector spaces, the dual space algebra_5e8002a6a49554c56f2d6061bf740617420e3787.png then has dimension

algebra_e02a79180b457b7927e5f035fe9711b005932645.png

as well.

Furthermore, algebra_72dbe25d162796389218b162ed2c6bf6a0711ac0.png form a basis for algebra_5e8002a6a49554c56f2d6061bf740617420e3787.png. Therefore we can write the elements in algebra_5e8002a6a49554c56f2d6061bf740617420e3787.png as

algebra_58e8edf1974882684e5c5c4342cadca7707ec008.png

algebra_796c4aae79f6c10e5d41b68166f3c6dc1c762c7d.png

First observe that if algebra_9c30dcc624147dc83c1eb29849267c0423ead04e.png, then algebra_a14d8487fae6909c522cdd676b01387e7c5d213d.png is fully defined by how it maps the basis elements algebra_1266bb0174a135a03fdf3f71599c1205468dffa8.png, and

algebra_c27c5ac6efc97c00d28930173758bfb8aafa187f.png

Then observe that if algebra_2fe1db7deaaa1772cfc2440934a2ca47f26e0d51.png, then

algebra_4cbe7d7ee1348e47db6a8bba0a38a245b8efcf17.png

Hence, we have a natural homomorphism which simply takes algebra_a14d8487fae6909c522cdd676b01387e7c5d213d.png, with coefficients algebra_a62055427e31bc4f71eba27b2b2386d964e1af20.png to the corresponding element algebra_09020c8265c9f042902b70f5085a2940c2f9104b.png with the same coefficients! That is, the isomorphism is given by

algebra_317b3fdf05c5ad294162c391d21f057b20edf9bb.png

Tensor algebra

Now we'll consider letting algebra_466c3284f3a97f982eade8cd7a1f2e64b840b768.png. We define tensor powers as

algebra_e59bfc2e85d01a2ec8fa4e2e628aac8708b38bd2.png

with algebra_4f71009961a91249a487bacafd0e4c7b351dcd37.png, algebra_8f35c4256d517a6fa745b6b2911b6b8c64342e27.png, etc. We can think of algebra_02c2f3334df94e0da650cfcbdf86c6467cdc6374.png as the dual vector space of k-multilinear forms on algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png.

Combining all tensor powers using the direct sum we have define the tensor algebra

algebra_a8bb1b8ee3b3fde930402ec3e5f3386d88e540fc.png

whose elements are finite sums

algebra_86310809f7ff03b8cd4fbcb54964a05d50afcaf0.png

of tensor products of vectors algebra_80155900b5b3fa43fc1aa5f39cfeded1c2149403.png.

The "multiplication" in algebra_a218173f83cae8b1995579c42c0cc544d45d99bc.png is defined by extending linearly the basic product

algebra_0b9012bf2ed1d5a58ebdcea96a3dda89c4be235d.png

The resulting algebra algebra_a218173f83cae8b1995579c42c0cc544d45d99bc.png is associative, but not commutative.

Two viewpoints

There are mainly due two useful ways of looking at tensors:

  1. Using Lemma lemma:homomorphisms-isomorphic-to-1-1-tensor-product, we may view algebra_c5bfaaa196662a4755b329b6c4c06626cf891b5d.png tensors as linear maps algebra_710ef6d88fddad483a76650b7fe13c73cd479508.png. In other words,

    algebra_19571d7d5fda929e24253845e7aae6df04860674.png

    • Furthermore, we can view algebra_2c226bf4f8ed3e1460da65439aac7c49cbefd0ed.png as the dual of algebra_0c214dcd764f714d2f5ce644dfe02be1dc51ed48.png, i.e. consider a algebra_e34e1166a0538fbe9967b8667c20460fcfe3e8da.png tensor as a "multilinear machine" which takes in algebra_ab524acaeebdc667a27bf814ed4b4b185a5f3c7b.png vectors and algebra_6ee30c172ceb70f3426abc02237d9f23445ae2f9.png co-vectors, and spits out a real number!
  2. By explicitly considering bases algebra_93868c7080638936e809863f0c473a2dfd0a4353.png of algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png and algebra_3802968e7ccb8e030d7dc17e89c2c26dffe5112d.png of algebra_b761c364e285a25aa55ec1ce6ccdd793a2418325.png, the algebra_c5bfaaa196662a4755b329b6c4c06626cf891b5d.png tensors are fully defined by how they map each combination of the basis elements. Therefore we can view tensors as multi-dimensional arrays!

Tensors (mainly as multidimensional arrays)

Let algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png be a vector space algebra_1c456d691bf8463d9af3a5188fe62bed5b466424.png where algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png is some field, algebra_8344b9c97f81222ed29999f04a6ac66ba7c7bbd7.png is addition in the vector space and algebra_c73ba8d9fee3e8c29911481fc0726828c3549c28.png is scalar-multiplication.

Then a tensor algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png is simply a linear map from some q-th Cartesian product of the dual space algebra_b761c364e285a25aa55ec1ce6ccdd793a2418325.png and some p-th Cartesian product of the vector space algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png to the reals algebra_ad02fc814a4ba9ef833f6acacd935d6845e92c41.png. In short:

algebra_453e5f643f9f14f22d8a5fbb89d9aa37de4a8e15.png

where algebra_51d6621265873ac7318e5266a819e786e89806f9.png denotes the (p, q) tensor-space on the vector space algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png, i.e. linear maps from Cartesian products of the vector space and it's dual space to a real number.

Tensors are geometric objects which describe linear relations between geometric vectors, scalars, and other tensors.

A tensor of type algebra_9b5ab2f12f92073f3a59aec48dd3f69d6e9809f6.png is an assignment of a multidimensional array

algebra_f4137cec8217204928cc7f0f08c405bad79db5b6.png

to each basis algebra_49d94a1b156c8e892bbf29e48b8ede47dfcbcbaa.png of an n-dimensional vector space such that, if we apply the change of basis

algebra_c698017e4d4b0347a0b350831dbd92819f635bad.png

Then the multi-dimensional array obeys the transformation law

algebra_875c52bda1c76fe16e2b6a88882dc2fb1989908c.png

We say the order of a tensor is algebra_5403f6bbface4889f05450c96efe5dfafd041d71.png if we require an n-dimensional array to describe the relation the tensor defines between the vector spaces.

Tensors are classified according to the number of contra-variant and co-variant indices, using the notation algebra_9b5ab2f12f92073f3a59aec48dd3f69d6e9809f6.png, where

  • algebra_b346b8eaf5913ed81b033dacc1d6568c142bbb4a.png is # of contra-variant indices
  • algebra_9974194a836f8c06efda1c78a55ab184c6c2d746.png is # of co-variant indices

Examples:

The tensor product takes two tensors, algebra_391ba25320bcd9d67b1952507e2350803f994ba7.png and algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png, and produces a new tensor, algebra_6bb380375d45f5a579829539aa9039eb1b559b49.png, whose order is the sum of the orders of the original tensors.

When described as multilinear maps, the tensor product simply multiplies the two tensors, i.e.

algebra_478b9e34e95a71fbe65617095e5d580ec9fe0e64.png

which again produces a map that is linear in its arguments.

On the components, this corresponds to multply the components of the two input tensors pairwise, i.e.

algebra_799390809b8866993abc2b5796c85dd4650413c4.png

where

  • algebra_391ba25320bcd9d67b1952507e2350803f994ba7.png is of type algebra_018c60620ad6d14d7f221890a6294351f15b7c5a.png
  • algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png is of type algebra_a1e9804845b91fe840ddb795f0e22586662cf531.png

Then the tensor product algebra_6bb380375d45f5a579829539aa9039eb1b559b49.png is of type algebra_87219004bb63fcf3e2a8fca5c93cdd24c29fcbd4.png.

Let algebra_6181c720dc1e1fd4dbf4ee689d1bd13c1333be3c.png be a basis of the vector space algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png with algebra_e5fcb072a5aa82d2e2f128cd71132542bf6e9afe.png, and algebra_8b0e8c70246c7c7c6648f2191676cdea55460e23.png be the basis of algebra_b761c364e285a25aa55ec1ce6ccdd793a2418325.png.

Then algebra_5000340a30b13348509869b1f279daa55ff7257e.png is a basis for the vector space of algebra_68c7762b1361ba0293ef587679cfa0ac95b1dc97.png over algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png, and so

algebra_39d7e612f3ef2a163414cf0d1dc9c73a1b894528.png

Given algebra_172eb7826e5562a7f7d713e7837b0439fd25ef67.png algebra_64785f2a493a7e6a25684a0c4b19905e60fbc20e.png, we define algebra_cb009232afad7abf2fd133b9d5bf21cc2aa86f8d.png algebra_0b457358a8f7a3c2a4348fc2cca6eacbee5c8dbc.png

algebra_e5021c3fe0d349d7dfef7068a32f7e700e4fd4b5.png

Components:

algebra_4dcf721139d52bf2e651fe5bcc8f3fc33b8d6516.png

A contraction is basis independent:

algebra_b1493f31d3915578a5372dbeb2fc3d14bfd38c69.png

algebra_f585b80eda59a8e1beec27a70cab1ca89b642ebb.png

Generalized to algebra_172eb7826e5562a7f7d713e7837b0439fd25ef67.png by summing over combinations of the indices, alternating sign if wanting anti-symmetric. See the general definition of a wedge product for more for example.

Examples
  • Linear maps - matrices

    A linear map is represented as a matrix, and we say this is a tensor of order 2, since it requires 2-dimensional array to describe the relation.

Clifford algebra

First we need the following definition:

Given a commutative ring algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png and algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png modules M$ and algebra_53f1e85ad97d85e1d44c21eeddd7b49e5c1cf97a.png, an algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png quadratic function algebra_aa45875592dca8c42175d80508685cdc5a3a50b8.png s.t.

  • (cube relation): For any algebra_a34ac6c88498c628e772466b24f9d3f7780201d8.png we have

    algebra_8ff0993715b4a8117f730efbb0bc4e678ff4b038.png

  • (homegenous of degree 2): For any algebra_b3194010d759b07b81a2fb8e290a917a84b09ab5.png and any algebra_4a6866e3855a59ae16bff6843a7c58d555ddf99e.png, we have

    algebra_656094acf7b173acc32a334ff19526e7a10b673b.png

A quadratic R-module is an algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png module algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png equipped with a quadratic form: an R-quadratic function on algebra_4041476b998bc5a6ca4d7b8f4880e883a5a433a8.png with values in algebra_76c425f1c1b4193275045ddb6cf6d9b72be71d87.png.

The Clifford algebra algebra_8084e19c6a561423826077bb081902d0ad36dc9f.png of a quadratic R-module algebra_f11df253a386ceab480c4f68097432338e1feeae.png can be defined as the quotient of the tensor algebra algebra_d82c43ba1107227063c47063aeccf91a955be2a8.png by the ideal generated by the relations algebra_7d8089d96b97505f5b0edf7401187991c4e3d176.png for all algebra_b3194010d759b07b81a2fb8e290a917a84b09ab5.png; that is

algebra_c48aa676b6d36a21717eaae4469e34ce0e7c9e6c.png

where algebra_fac4953b65d5acce23bb990e75d9b9c21fdb1358.png is the ideal generated by algebra_7d8089d96b97505f5b0edf7401187991c4e3d176.png.

In the case we're working with a vector space algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png (instead of a module), we have

algebra_b70b2f440b6d7525ccc65aca23595ac6a82bfd5b.png

Since the tensor algebra algebra_a218173f83cae8b1995579c42c0cc544d45d99bc.png is naturally algebra_90b9284372e914cdce851c9a4e2a5260e6bb7cde.png graded, the Clifford algebra algebra_dc7f5800bec5a5df06665bf458a4b263ded68abf.png is naturally algebra_02e4160fbdf6440be5e79218d3a6cf638031f371.png graded.

Examples

Exterior / Grassman algebra

  • Consider a Clifford algebra algebra_dc7f5800bec5a5df06665bf458a4b263ded68abf.png generated by the quadratic form

    algebra_c5a0234c0a6adb43413dc305d836b759603cf553.png

    i.e. identically zero.

  • This apparently gives you the Exterior algebra over algebra_b339a910bb6007d4d79e9c904b54a684fcf79e70.png
    • A bit confused as to why