# MA251 Algebra 1: Advanced Linear Algebra

Term(s): Term 1

Status for Mathematics students: Core for Maths

Commitment: 30 one-hour lectures plus three assignments

Assessment: 85% by 2-hour examination, 15% coursework

Formal registration prerequisites: None

Assumed knowledge: Knowledge of Vector spaces and matrices from MA106 Linear Algebra. In particular, understanding change of basis matrices, eigenvalues and eigenvectors, elementary row and column operations and diagonalisation of matrices

Useful background: Group theory from MA136 Introduction to Abstract Algebra especially abelian groups

Synergies: The following modules go well with this module: MA249 Algebra 2: Groups and Rings

Leads to: The following modules have this module listed as assumed knowledge or useful background:

Content: This module is a continuation of MA106 Linear Algebra. In that course we studied conditions under which a matrix is similar to a diagonal matrix, but we did not develop methods for testing whether two general matrices are similar. Our first aim is to fill this gap for matrices over ${\mathbb C}$. Not all matrices are similar to a diagonal matrix, but they are all similar to one in Jordan canonical form; that is, to a matrix which is almost diagonal, but may have some entries equal to 1 on the superdiagonal.

We next study quadratic forms. A quadratic form is a homogeneous quadratic expression $\sum a_{ii}x_ix_j$ in several variables. Quadratic forms occur in geometry as the equation of a quadratic cone, or as the leading term of the equation of a plane conic or a quadric hypersurface. By a change of coordinates, we can always write $q(x)$ in the diagonal form $\sum a_ix_i^2$. . For a quadratic form over ${\mathbb R}$, the number of positive or negative diagonal coefficients $a_i$is an invariant of the quadratic form which is very important in applications.

Finally, we study matrices over the integers ${\mathbb Z}$, and investigate what happens when we restrict methods of linear algebra, such as elementary row and column operations, to operations over ${\mathbb Z}$. This leads, perhaps unexpectedly, to a complete classification of finitely generated abelian groups.

Aims: To develop further and to continue the study of linear algebra, which was begun in Year 1.

To point out and briefly discuss applications of the techniques developed to other branches of mathematics, physics, etc.

Objectives: By the end of the module students should be familiar with: the theory and computation of the the Jordan canonical form of matrices and linear maps; bilinear forms, quadratic forms, and choosing canonical bases for these; the theory and computation of the Smith normal form for matrices over the integers and its application to finitely generated abelian groups.

Books:

P M Cohn, Algebra, Vol. 1, Wiley

I N Herstein, Topics in Algebra, Wiley.

Neither is essential, but are a good idea if you are intending to study further algebra modules.