# Course Content

The purpose of this course is to give you an introduction to Linear Algebra, which is one of the most important technical tools both in Pure and Applied Maths. The topics covered will be
• Systems of simultaneous linear equations. Examples.
• Gauss--Jordan elimination.
• Row echelon form for a rectangular matrix.
• Fredholm's alternative. Applications.
• Computing the inverse matrix using row operations.
• Odd and even permutations. Determinants.
• Row and column operations on determinants. Determinant of the transpose matrix.
• Minors. Cofactors. Adjoint matrix. Computing the inverse matrix using determinants.
• Cramer's rule for systems with the same number of equations and unknowns.
• Coordinate vector space. Ranks. Maximal size of nonzero minors is equal to the rank.
• Fields: rationals, reals, and complex.
• Abstract vector spaces.
• Linear independence and completeness. Exchange lemma.
• Bases and dimensions. Subspaces.
• Linear operators. Matrices.
• Change of basis. Transition matrices. Similar matrices define the same linear operator in different bases.
• Characteristic polynomials. Eigenvalues and eigenvectors.
• Diagonalisation in the case when the characteristic polynomial has no multiple roots.
• Cayley--Hamilton theorem. Minimal polynomial of a linear operator. Examples (operators with A2=A).
• Invariant subspaces. Direct sums.
• Normal form for a nilpotent operator. Jordan normal form (Jordan Decomposition Theorem).
• Applications: computing functions of matrices, solving differential equations, finding closed expressions for recursively defined sequences.
• Orthonormal bases; Gram--Schmidt orthogonalisation.
• Orthogonal complements and orthogonal direct sums. Bessel's inequality.
• Bilinear and quadratic forms. Sylvester's criterion. The law of inertia.
• Spectral Theorem for symmetric operators.
• Complex Hilbert spaces.
• Two commuting linear operators have a common eigenvector.
• Spectral Theorem for normal operators.

# Materials

The materials on this webpage are in two different formats: PDF and PS (Postscript). PDF files are handled, for example, by Adobe Acrobat Reader. It makes sense to learn how to handle PS files as well. I suggest some public-domain software that opens these files: Ghostview, a user-friendly interface to Ghostscript (so you need both of these).

## Exam materials: sample papers, actual papers, solutions

A sample paper for the Christmas exam [PS] [PDF]
A sample paper for the Easter exam [PS] [PDF]
Christmas half-exam [PS] [PDF]
Solutions to the Christmas paper [PS] [PDF]
Easter half-exam [PS] [PDF]
Solutions to the Easter paper [PS] [PDF]
A sample exam paper [PS] [PDF]
Solutions to the sample exam paper [PS] [PDF]
Final exam paper: [PS] [PDF]
Solutions to the final exam paper: [PS] [PDF]
A sample supplemental paper [PS] [PDF]
Solutions to the sample supplemental paper [PS] [PDF]

## Textbooks

There will be no lecture notes for this course, so you are encouraged to take notes during the lectures - it takes effort but is really helpful. There are some books which you might find helpful. None of them fit exactly the content and order of presentation of the course. Have a glance at Anton/Rorres' Elementary Linear Algebra (applications version), Strang's Linear Algebra and its Applications, and, the last but not the least, Gelfand's Lectures on Linear Algebra. You can also attempt problems from Linear Algebra Problem Book by Paul Halmos.

## Handouts

Handout 1 (xerox copy of some pages from Gelfand): "Vector spaces and isomorphisms"
Handout 2 (xerox copy of some pages from Gelfand): "Linear operators (transformations), eigenvalues, eigenvectors"
Handout 3 (not from Gelfand): "Linear operators on a finite-dimensional vector space: a brief HOWTO" [PS] [PDF]
Handout 4 (not from Gelfand): "Jordan normal form theorem" [PS] [PDF]
Handout 5 (not from Gelfand): "Orthonormal bases, orthogonal complements, and orthgonal direct sums" [PS] [PDF]
Handout 6 (xerox copy of some pages from Gelfand): "Euclidean spaces and bilinear forms (orthogonal bases, canonical forms, law of inertia, Sylvester's criterion)"
Handout 7 (xerox copy of some pages from Gelfand): "Spectral theorem for symmetric operators. Complex Hilbert space and spectral theorem for normal operators"
Handout 8 (not from Gelfand): "Several problems in Linear Algebra" [PS] [PDF]

## Assessment etc.

You will get home assignments each week. There will be two half-exams (90 minutes each) on the weeks after the end of Michaelmas and Hilary terms, and a 3-hour exam in the end of the year. Your final grade will be maximal of 100% final exam, and 70% of final exam plus 15% of home assignments grade plus 15% of the best of two half-exam results. On some days we shall have tutorial classes --- mostly to discuss questions similar to ones from recent assignments, but also with an opportunity for you to ask questions on the current assignment (if needed), and tell others your solutions for the previous assignments.