References cited:
[S] G. Strang: Linear Algebra and Its Applications, (3rd ed)
[W] L.W. Johnson, Riess, Arnold: Introduction to Linear Algebra
[L] P. Lax: Linear Algebra
[Sh] G. Shilov, Linear Algebra

Syllabus (in essence):

I. VECTOR SPACES
Definition. Subspaces
(references: [W] 4.1-4.5, [S] 2.1, 2.3, [L] Ch.1)
Spanning sets, Linear independence, Bases and coordinates, Dimension
(references: [W] 4.1-4.5, [S] 2.1, 2.3, [L] Ch.1)

0. DETERMINANTS
(references: [Sh] Ch.1, [S] Ch.4)

II. 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
(references: [W] 4.7-4.9, [S]: 2.4, 2.6, [L]: Ch.3, 4)
The solution of m equations in n unknowns
(ref: [S] 2.2, [W] Ch.1)

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. 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

V. SELF-ADJOINT MATRICES
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

VI. OTHER APPLICATIONS
Powers of a matrix, functions of matrices, linear differential equations
Difference equations
Markov processes