Syllabus (in essence):

I. VECTOR SPACES
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

Adjoint of a matrix
Hermitian matrices
Spectral theorem
Triangularization via unitary similarity transformation
Diagonalization of normal matrices
Positive de finite 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