Advanced Linear Algebra (Applied MS)

Preparatory Course: Math 5023

1.

Prerequisites: The arithmetic of matrices, solving systems of equations using matrix techniques, the matrix of a linear transformation with respect to a given basis, computing determinants and eigenvalues, and using similar matrices to find powers and roots of a given matrix.

2.

Vector spaces: subspaces, basis dimension, quotient spaces, field, independence, spanning, scalar product, orthonormal basis, and Gram-Schmidt

3.

Linear transformations: examples of linear transformations, kernel, image, rank, invertibility, diagonalization, reduced echelon form, matrix of a linear transformation with respect to given bases, similarity, and classical adjoint, and extending a linear transformation defined on a subspace to the vector space by telling what it does to a basis

4.

Determinants: permutation definition and elementary properties

5.

Canonical forms: eigenvectors, eigenvalues, characteristic polynomial, minimal polynomial, symmetric matrix, direct sum decomposition, invariant subspaces, and elementary divisors, Jordan and the Rational canonical forms of a matrix, and the Cayley-Hamilton theorem

6.

Dual spaces: dual space, dual basis, and adjoint

REFERENCES: C.W. Curtis, Linear Algebra:An Introductory Approach; Seymour Lipschutz, Schaum's Outline of Theory and Problems of Linear Algebra