Vector spaces

Subspaces

Spanning sets

Linear independence

Bases and coordinates

Dimension

II. LINEAR OPERATORS

Linear operators

Representation as matrices

Null space (kernel) and range space

Operations with linear transformations

Invertible operators

Isomorphism

Change of basis, the transition matrix, similarity of matrices

The solution of m equations in n unknowns

III. INNER PRODUCT SPACES

Inner product, norm, metric

Orthonormal bases

Gram-Schmidt orthogonalization process

Orthogonal matrices

Right and left inverses

Least squares approximation, Bessel's inequality

The four fundamental subspaces of a matrix

The Fredholm alternative

Intersection and sum of two vector space

IV. THE DUAL SPACE (covectors)

Linear and multilinear functionals.

V. EIGENVALUES AND EIGENVECTORS (SPECTRAL THEORY)

Invariant subspaces, commuting operators

Eigenvalues and eigenvectors

Determinant (signed volume, multiplicative property), trace

Eigenvector basis

Diagonalizable matrices

Solutions of linear differential equations using eigenvalues and eigenvectors

VI. SELF-ADJOINT MATRICES

Adjoint of a matrix

Hermitian matrices

Spectral theorem

Triangularization via unitary similarity transformation

Diagonalization of normal matrices

Positive definite matrices

Quadratic forms and the generalized eigenvalue problem

Extremization with linear constraints

Rayleigh quotient

Singular value decomposition of a rectangular matrix

Pseudo-inverse of a rectangular matrix

Small oscillations

VII. OTHER APPLICATIONS

Depending on the interest of the students and the remaining time, we may also discuss

Vandermonde determinants

Circulant matrices, Toeplitz matrices

An introduction to tensors

Powers of a matrix, functions of matrices, linear differential equations

Difference equations

Markov processes