R1 Final Review, Part 1: Linear Algebra

• (Lec. 6) Examples: , , (characteristic : ); , prime (characteristic ).

• Examples: , , …
• Subspace: closed under and .

• (Axler ch. 2) Say are linearly independent if for all .
• (Lec. 7) Basis = linearly independent vectors which span : then , , is an isomorphism.

• Any linearly independent set can be completed to a basis.

• (Lec. 8) Then is represented in the basis by the column vector :
• Change of basis: , i.e. ; then for , .

• basis of (finite-dimensional) dual basis of such that
• , , is an isomorphism if (injective if ).
• The annihilator of is ; .
• The transpose of is , . , if ; .

• , , is surjective with kernel .
• factors through (there exists with ) iff .

(Axler ch. 5) is an invariant subspace for if .

• Examples: , ; eigenspaces .

(Lec. 10) 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.

(Lec. 11) 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 .

• (Lec. 12) 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 .

(Handout) Categories have objects, and morphisms for all , with operation = composition.

• Axioms: for all there exists with ; associativity .
• Examples: sets, groups, vector spaces over .

(Lec. 14) 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; Lec. 14) 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 .

(Lec. 15) 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.

(Lec. 16) is self-adjoint if .

• (Lec. 17) 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; Lec. 18) 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 , .

(Lec. 19) Multilinear maps linear maps .

contains subspaces

• = symmetric tensors ( symmetric multilinear maps): ;
• = exterior powers: alternating tensors, .

(Lec. 20) 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.

• (Lec. 21; 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 .