Aims

• To develop the theory of vector spaces by extending the ideas presented in Core Algebra.
• To characterise matrices in terms of their Jordan normal form, and hence prescribe when they can be diagonalised and when they cannot.
• To describe the main properties of normal matrices (eg. symmetric, hermitian, orthogonal, unitary).
• To find some decompositions of matrices.
• To develop the theory of quadratic forms.

Learning objectives

At the end of the module you should:

• Be comfortable with the axiomatic definition of a vector space.
• Deduce simple properties of vector spaces from the axioms.
• Represent linear maps by matrices.
• Understand the concepts of inner product and inner product space.
• Understand the main properties of hermitian, unitary and normal matrices and linear maps.
• Understand why the Cayley-Hamilton Theorem is true, and find the algebraic, minimal and geometric multiplicities of eigenvalues of a square matrix.
• Understand the Jordan normal form of a square matrix.
• Decide whether a given matrix can be diagonalised.
• Understand some bilinear, quadratic and hermitian forms and be able to diagonalise them.

Syllabus.

SPRING TERM

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.

Recommended texts

• *** R B J T Allenby, Linear Algebra, Arnold (S 2.897 ALL). 
• R Kaye and R Wilson, Linear Algebra, OUP (S 2.897 KAY).
• D C Lay, Linear Algebra and its applications, Addison Wesley (S 2.897 LAY).
• J. B. Fraleigh and R. A. Beauregard, Linear Algebra, Addison Wesley (S 2.897 FRA).
• P. R. Halmos, Linear Algebra Problem Book, MAA ( S 2.897 HAL).

Teaching

• 3 lectures and 1 seminar per week in Weeks 2-10 of the Spring term and Weeks 1-3 of the Summer term
• 2 revision classes Week 4 of the Summer term.

Assessment

Three hour closed examination.

Elective information

This module is not available as an elective.

Prerequisites

• All taught in Core Algebra