Course category:2nd year
At the end of the module you should:
Vector spaces. Definition of field and characteristic; vector spaces over a general field; bases; dimension; subspaces; intersection, sum, direct sums and complement.
Homomorphisms. Linear maps, homomorphisms, endomorphisms, isomorphisms, automorphism; image and kernel of a map; rank-nullity theorem.
Quotient spaces and tensor products. Quotient spaces; tensor products defined both as a quotient and via bases. (if time permits)
Matrices. Representing homomorphisms by matrices; composition of two homomorphisms; change of basis; similar matrices; rank-nullity for matrices.
Inverses and determinants. Inverse, trace and determinant of a square matrix, including Laplace expansion by cofactors and adjugate.
Eigenvalues. Eigenvalues and eigenvectors; the characteristic polynomial; Cayley-Hamilton theorem; minimal polynomial; the algebraic, geometric and minimal multiplicity of an eigenvalue.
Diagonalizability. When is a matrix diagonalizable (characterize in terms of eigenvalue multiplicities etc.); the Jordan normal form (over the complex numbers only).
Dual spaces. Algebraic dual of a finite dimensional space (including examples from inner product spaces); dual maps; double dual and its canonical isomorphism for finite dimensional spaces; example of double dual in an infinite dimensional space (briefly if time permits).
Inner product spaces. Real and complex inner products; norm and distance coming from an inner product; metric spaces (briefly if time permits)
Orthogonality. Gram-Schmidt process; projections; orthogonal complement of an inner-product subspace.
Spectral theorems. Normal matrices (eg., symmetric, orthogonal, hermitian, unitary); Schur triangularization (any complex matrix is unitarily similar to an upper triangular one, needs Gram-Schmidt to construct an orthonormal basis); the spectral theorem (a matrix is unitarily similar to a diagonal matrix if and only if it is normal); spectral theory for self-adjoint maps and isometries on inner product spaces (direct proofs, not via normal matrices) (if time permits)
Plus a selection of topics from:
Matrix Groups. Special unitary, special orthogonal and symplectic groups, and their invariant bilinear forms, with some physical examples.
Quadratic forms. Bilinear, sesquilinear, quadratic and hermitian-quadratic forms. Rank and signature; Sylvester's law of inertia.
Matrix decompositions. Singular value decomposition; polar decomposition; QR decomposition; other matrix decompositions.
Three hour closed examination.
This module is not available as an elective.
Department of Mathematics, University of York, Heslington, York, UK. YO10 5DD