# M1: Linear Algebra II - Material for the year 2019-2020

## Primary tabs

8 lectures

To follow.

Students will:

(i) understand the elementary theory of determinants;

(ii) understand the beginnings of the theory of eigenvectors and eigenvalues and appreciate the applications of diagonalizability.

(iii) understand the Spectral Theory for real symmetric matrices, and appreciate the geometric importance of an orthogonal change of variable.

See the examinable syllabus.

Introduction to determinant of a square matrix: existence and uniqueness. Proof of existence by induction. Proof of uniqueness by deriving explicit formula from the properties of the determinant. Permutation matrices. (No general discussion of permutations). Basic properties of determinant, relation to volume. Multiplicativity of the determinant, computation by row operations.

Determinants and linear transformations: definition of the determinant of a linear transformation, multiplicativity, invertibility and the determinant.

Eigenvectors and eigenvalues, the characteristic polynomial, trace. Eigenvectors for distinct eigenvalues are linearly independent. Discussion of diagonalisation. Examples. Eigenspaces, geometric and algebraic multiplicity of eigenvalues. Eigenspaces form a direct sum.

Gram-Schmidt procedure. Spectral theorem for real symmetric matrices. Quadratic forms and real symmetric matrices. Application of the spectral theorem to putting quadrics into normal form by orthogonal transformations and translations. Statement of classification of orthogonal transformations.

- T. S. Blyth and E. F. Robertson,
*Basic Linear Algebra*(Springer, London, 2nd edition 2002). - C. W. Curtis,
*Linear Algebra -- An Introductory Approach*(Springer, New York, 4th edition, reprinted 1994). - R. B. J. T. Allenby,
*Linear Algebra*(Arnold, London, 1995). - D. A. Towers,
*A Guide to Linear Algebra*(Macmillan, Basingstoke 1988). - S. Lang,
*Linear Algebra*(Springer, London, Third Edition, 1987). - R. Earl,
*Towards Higher Mathematics - A Companion*(Cambridge University Press, Cambridge, 2017)