Synopsis for Linear Algebra II

Learning Outcomes

Students will:

1. understand the elementary theory of determinants;
2. understand the beginnings of the theory of eigenvectors and eigenvalues and appreciate the applications of diagonalizability.
3. understand the Spectral Theory for real symmetric matrices, and appreciate the geometric importance of an orthogonal change of variable.

Synopsis

Introduction to determinant of a square matrix: existence and uniqueness and relation to volume. Proof of existence by induction. Basic properties, computation by row operations.

Determinants and linear transformations: multiplicativity of the determinant, definition of the determinant of a linear transformation. Invertibility and the determinant. Permutation matrices and explicit formula for the determinant deduced from properties of determinant. (No general discussion of permutations).

Eigenvectors and eigenvalues, the characteristic polynomial. Trace. Proof that eigenspaces form a direct sum. Examples. Discussion of diagonalisation. Geometric and algebraic multiplicity of eigenvalues.

Gram-Schmidt procedure.

Spectral theorem for real symmetric matrices. Matrix realisation of bilinear maps given a basis and application to orthogonal transformation of quadrics into normal form. Statement of classification of orthogonal transformations.

1. T. S.  Blyth and E. F.  Robertson, Basic Linear Algebra (Springer, London 1998).
2. C. W. Curtis, Linear Algebra – An Introductory Approach (Springer, New York, 4th edition, reprinted 1994).
3. R. B. J. T.  Allenby, Linear Algebra (Arnold, London, 1995).
4. D. A.  Towers, A Guide to Linear Algebra (Macmillan, Basingstoke 1988).
5. S. Lang, Linear Algebra (Springer, London, Third Edition, 1987).