Midterm 2 Review: Linear Algebra and Group Theory — Studies in Algebra and Group Theory

Part 1: Linear algebra

Note (Already on Midterm 1).

A field is a set with two operations: is an abelian group with identity ; is an abelian group with identity ; distributive law .
- Examples: , , (characteristic : ); , prime (characteristic ).
(Axler ch. 1) A vector space over is a set with addition ( an abelian group, ) and scalar multiplication , associative and distributive.
- Examples: , , …
- Subspace: closed under and .
; say span if .
- (Axler ch. 2) Say are linearly independent if for all .
- Basis = linearly independent vectors which span : then , , is an isomorphism.
All bases of have the same cardinality .
- Any linearly independent set can be completed to a basis.
= linear maps : , . This is a vector space.
(Axler ch. 3) Given bases of and of , represent by the column , and by the matrix whose columns represent in the basis : .
- Then is represented in the basis by the column vector :
- Change of basis: , i.e. ; then for , .
is a direct sum decomposition if i.e. , , is an isomorphism.
finite-dimensional iff and .
Dim/rank formula: finite-dimensional, .
There exist bases of , of such that where the first rows correspond to and the last columns to .
Dual: .
- basis of (finite-dimensional) dual basis of such that
- , , is an isomorphism if (injective if ).
- The annihilator of is ; .
- The transpose of is , . , if ; .
Quotient: subspace is a vector space.
- , , is surjective with kernel .
- factors through (there exists with ) iff .

(Axler ch. 5) is an invariant subspace for if .
- Examples: , ; eigenspaces .
If with invariant for , then there is a basis where is block diagonal: .
- A basis of eigenvectors , , diagonalizable: .
Eigenvectors of for distinct eigenvalues are linearly independent.
If is algebraically closed (e.g. ) then any linear operator has an eigenvector.
- Corollary: there exists a basis in which is upper triangular, .
- is an eigenvalue of is not invertible appears on the diagonal in a triangular matrix representing .

(Axler ch. 8) The generalized kernel for all large (e.g. ).
- is nilpotent if ; . There exists a basis such that is block diagonal with blocks .
The generalized eigenspaces are linearly independent invariant subspaces.
If is algebraically closed then = the direct sum of the generalized eigenspaces of .
- This gives the Jordan normal form: is block diagonal with blocks ( diagonalizable all blocks have size ).
Characteristic polynomial of : , .
- Minimal polynomial: , = size of the largest Jordan block in .
- iff . In particular . diagonalizable for all .
Over , need not have eigenvectors, but by considering and , any real operator has an invariant subspace of dimension (eigenvector!) or .

Note (Not on midterm).

(Handout) Categories have objects, and morphisms for all , with operation = composition.
- Axioms: for all there exists with ; associativity .
- Examples: sets, groups, vector spaces over .
A functor assigns
- to each , ;
- to each , ,
such that and . (Contravariant functors = reverse direction of morphisms.)
A natural transformation between functors = for each , such that for all , commutes.

(Axler ch. 9A) A bilinear form on is , linear in each input: , , etc.
- is symmetric if , skew-symmetric if .
(isomorphism of vector spaces), .
- ; is nondegenerate if is an isomorphism.
In a basis of , is represented by a matrix .
- If , are represented by column vectors , then .
The orthogonal of for is .
- If is nondegenerate then .
- (Axler ch. 6) If is an inner product then and .
A real inner product is a symmetric definite positive bilinear form ( for all ).
- Cauchy–Schwarz inequality: .
- Over , we consider Hermitian inner products , not quite bilinear: ; we require Hermitian-symmetric, , and definite positive, for all . The map induced by such is -antilinear: .
Every finite-dimensional inner product space (over or ) has an orthonormal basis such that (build by induction, e.g. using Gram–Schmidt).
(Axler ch. 7) Let be an inner product space (over or ), a linear operator.
- The adjoint operator satisfies for all . (It corresponds to the transpose of via ; over : the complex conjugate of .)
- In an orthonormal basis, (real case) or (complex Hermitian case).
- and vice versa.
is self-adjoint if .
- is orthogonal (unitary over ) if , i.e. for all ( maps an orthonormal basis to an orthonormal basis).
If is invariant under a self-adjoint/orthogonal/unitary operator then so is . Spectral theorem (real and complex versions):
- If is self-adjoint then is diagonalizable, with real eigenvalues, and can be diagonalized in an orthonormal basis.
- If is orthogonal for a real inner product, then is a direct sum of orthogonal invariant subspaces of dimension or , with acting by on the 1-dimensional pieces and by rotations on the 2-dimensional pieces.
- If is unitary for a Hermitian inner product, then is diagonalizable in an orthonormal basis, with eigenvalues . More generally, if (Hermitian) is normal, i.e. , then is diagonalizable in an orthonormal basis.
Besides inner products, one can also consider arbitrary nondegenerate symmetric bilinear forms (without assuming positivity); e.g. over (resp. ) there exists an orthogonal basis such that (resp. ); or skew-symmetric bilinear forms.
(Handout) Tensor product: is a vector space, with a bilinear map , , such that bilinear maps correspond to linear maps ().
- Elements of are finite linear combinations .
- If is a basis of and a basis of , then is a basis of .
, by mapping to .
The trace can be defined by , .

Note (Not on midterm).

Multilinear maps linear maps .
contains subspaces
- = symmetric tensors ( symmetric multilinear maps): ;
- = exterior powers: alternating tensors, .
If then has dimension ; for , is multiplication by a scalar, the determinant .
The theory of modules over a ring (elements need not have multiplicative inverses) is more complicated than that of vector spaces.
- Finitely generated modules need not have a basis; those that do are called free.

-modules abelian groups.
- (Parts of Artin ch. 14) Every finitely generated -module with generators is a quotient of (, ), and is itself a free module, i.e. there exists such that .
- Via linear algebra over , one finds: every finitely generated abelian group is for some . (Recall if .)

Part 2: Group theory

Note (Already on Midterm 1).

(Artin ch. 2) A group is a set with an operation such that
1. there exists , the identity, such that for all ;
2. for all there exists an inverse such that ;
3. associativity: for all .
A group is abelian if is commutative ( for all ).
Examples: , , , the symmetric group ; , etc.; products , , …
Like sets, groups can be finite (, , …), countable (, , , …) or uncountable (, …).
is a subgroup if ; ; . divides .
- subgroups of is a subgroup of .
- All subgroups of are for some .
A homomorphism is a map such that for all ().
- Isomorphism = bijective homomorphism; automorphism = isomorphism . is a group.
The kernel of : , a subgroup of . injective .
- The image of : , a subgroup of . surjective .
Given , , , is a homomorphism with , the subgroup generated by .
- , where = order of = .
- Hence the cyclic group is if has order , and if has infinite order.
- ( generate if every element of is a product of the and their inverses.)
A subgroup determines an equivalence relation (axioms: ; ; and ): iff , whose equivalence classes are the (left) cosets .
- The quotient set: . The index of : if finite.
If is finite: subgroup divides ; ; prime .
A subgroup is normal for all (left cosets = right cosets) for all (conjugate subgroups).
The operation makes a group iff is a normal subgroup.
For every homomorphism , is a normal subgroup of , and .
- If is surjective, we have an exact sequence ( injective, surjective, ).
- Examples: ( normal); ( iff ); .
- A homomorphism factors through iff .
is simple if its only normal subgroups are and itself. Examples: , prime; , .
Example: the center is a normal subgroup (abelian: ).

Example: the commutator subgroup , where , is normal, and (abelianization) is the largest abelian quotient of : every with abelian factors as .
(Artin 14.7) Every finitely generated abelian group is for some . (Recall if .)
(Artin 6.7–6.12) Group actions: a -action on a set is , , such that for all and ( a homomorphism ).
- Faithful if is injective; transitive if for all there exists such that (i.e. one orbit).
The orbit of is . These form a partition orbits.
- The stabilizer of is , a subgroup of .
- Elements in the same orbit have conjugate stabilizer subgroups: .
- Orbit-stabilizer: if , then , , is a bijection; in particular .
Burnside’s lemma (, finite): let be the fixed points of ; then .
(Artin ch. 7) acts on itself by left multiplication. This gives , hence every finite group is isomorphic to a subgroup of , .
acts on itself by conjugation: acts by .
- Orbits = conjugacy classes; , the centralizer of .
- Hence , where for each conjugacy class divides (the class equation of ).
For -groups (), the class equation (the number of conjugacy classes of size ). Hence:
- , prime is abelian ( or ).
- 5 isomorphism classes of groups of order 8: , , , , the quaternion group.
finite subgroup by considering the action of on its poles (unit vectors along rotation axes), one of , (regular -gon), (tetrahedron), (cube), (dodecahedron/icosahedron).
The symmetric group is generated by transpositions , in fact by .
For all there is a unique decomposition of as a product of disjoint cycles .
- are in the same conjugacy class iff they have the same cycle lengths.
The alternating group .
- A conjugacy class in which consists of even permutations is either 1 or 2 conjugacy classes in ; it splits into 2 iff the centralizer ( the cycle lengths of are all odd and distinct).
is simple for ( isn’t: is normal in and ).

Midterm review problems

This is a fairly arbitrary selection of problems from the textbook that may be good for checking your understanding of the concepts. I also recommend going over the homework assignments (not worrying about any problems that were particularly tricky or lengthy).

Problem numbers in Axler refer to the 4th edition.

Midterm 1

Axler chapter 2: 2A.12; 2B.8; 2C.8, 2C.15, 2C.19
Axler chapter 3: 3B.3, 3B.5, 3B.19, 3B.22, 3B.24, 3B.25, 3B.27; 3C.2, 3C.7; 3D.5, 3D.10, 3D.17; 3E.1, 3E.15; 3F.20, 3F.22, 3F.24, 3F.25 (Note: Axler uses the notation for the field; “null” for “Ker” and “range” for “Im”; for ; for , for .)
Axler chapter 5: 5A.7, 5A.13, 5A.24, 5A.33; 5B.20; 5C.10; 5D.3, 5D.4; 5E.3
Artin chapter 2: 4.5, 4.7, 4.9, 4.10; 5.1, 5.2; 6.3, 6.10; 8.6, 8.7, 8.9, 8.10; 10.2; 11.1, 11.4; 12.1, 12.2; M.2, M.9, M.10 (A couple of these are a bit more involved than what is intended for problems on the midterm.)
Artin chapter 3: 1.9; 3.1; 4.5; 5.3.
Artin chapter 4: 4.2, 4.6.

Midterm 2

(Problems marked * are a bit more involved than what is intended for the midterm.)

Axler chapter 6: 6A.2; 6B.6*, 6B.15, 6B.21*; 6C.7, 6C.10*, 6C.12 (Note for 6C: Axler denotes by the orthogonal projection onto , see 6.55)
Axler chapter 7: 7A.4, 7A.5, 7A.6, 7A.23, 7A.27 (assume self-adjoint rather than normal); 7B.10, 7B.14, 7B.21, 7B.22, 7B.23*; 7C.19; 7D.8
Axler chapter 8: 8A.2, 8A.22; 8B.2, 8B.9; 8C.4, 8C.7; 8D.1, 8D.2
Artin chapter 4: 4.2, 4.6; 6.4, 6.8; 7.3, 7.6, 7.7; M.2*, M.7*, M.10*
Artin chapter 5: 2.3*, M.3, M.7*
Artin chapter 6: 4.2*, 4.3; 7.1, 7.6, 7.10, 7.11*; 8.4; 9.2, 9.4; 10.2, 10.3*; 11.1, 11.3, 11.5, 11.7*; M.3, M.4*
Artin chapter 7: 1.2; 2.2, 2.3*, 2.7, 2.13*, 2.17*; 3.2, 3.3; 4.2, 4.9*; 5.2, 5.3, 5.7, 5.12; M.6, M.7.
Artin chapter 8: 2.2; 4.2, 4.3, 4.10, 4.12; 5.2, 5.3, 5.4*; 6.2, 6.19; M.4*, M.14*
Artin chapter 14: 7.2, 7.5, 7.9*