Term	Michaelmas Term 2008  (14 lectures)

Overview

Linear algebra pervades and is fundamental to geometry (from which it originally arose), algebra, analysis, applied mathematics, statistics - indeed most of mathematics. Vector spaces will usually be discussed over the field of real numbers or complex numbers and also as a general field

Learning outcomes

Students will

1. understand the general concepts of a vector space, a subspace, linear dependence and independence, spanning sets and bases.
2. have an understanding of matrices and of their applications to the algorithmic solution of systems of linear equations and to their representation of linear transformations of vector spaces.

Synopsis

Algebra of matrices.

Vector spaces over the real numbers and more generally over field ; subspaces. Linear dependence and linear independence. The span of a (finite) set of vectors; spanning sets. Examples.

Definition of bases; reduction of a spanning set and extension of a linearly independent set to a basis; proof that all bases have the same size. Dimension of the space. Co-ordinates with respect to a basis.

Sums and intersections of subspaces; formula for the dimension of the sum.

Linear transformations from one (real) vector space to another. The image and kernel of a linear transformation. The Rank-Nullity Theorem. Applications.

The matrix representation of a linear transformation with respect to fixed bases; change of basis and co-ordinate systems. Composition of transformations and product of matrices.

Elementary row operations on matrices; echelon form and row-reduction. Matrix representation of a system of linear equations. Invariance of the row space under row operations; row rank.

Significance of image, kernel, rank and nullity for systems of linear equations including geometrical examples. Solution by Gaussian elimination. Bases of solution space of homogeneous equations. Applications to finding bases of vector spaces.

Invertible matrices; use of row operations to decide invertibility and to calculate inverse.

Column space and column rank. Equality of row rank and column rank.

Reading list

• C. W. Curtis, Linear Algebra - An Introductory Approach (Springer, 4th edition, reprinted 1994).
• R. B. J. T. �Allenby, Linear Algebra (Arnold, 1995).
• T. S. �Blyth and E. F. �Robertson, Basic Linear Algebra (Springer, 1998).
• D. A. �Towers, A Guide to Linear Algebra (Macmillan, 1988).
• D. T. �Finkbeiner, Elements of Linear Algebra (Freeman, 1972). [Out of print, but available in many libraries]
• B. Seymour Lipschutz, Marc Lipson, Linear Algebra (McGraw Hill, Third Edition 2001).
• R. B. J. T. Allenby, Rings, Fields and Groups (Edward Arnold, Second Edition, 1999). [Out of print, but available in many libraries also via Amazon.]

Taking our courses

This form is not to be used by students studying for a degree in the Department of Computer Science, or for Visiting Students who are registered for Computer Science courses

Other matriculated University of Oxford students who are interested in taking this, or other, courses in the Department of Computer Science, must complete this online form by 17.00 on Friday of 0th week of term in which the course is taught. Late requests, and requests sent by email, will not be considered. All requests must be approved by the relevant Computer Science departmental committee and can only be submitted using this form.