[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

Vector spaces

Subspaces

Spanning sets

Linear independence

Bases and coordinates

Dimension

(references: [W] 4.1-4.5, [S] 2.1, 2.3, [L] Ch.1)

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

Eigenspaces

Generalized eigenvectors and eigenspaces

Rank of a matrix

Solutions of linear differential equations using eigenvalues and eigenvectors

V. OTHER APPLICATIONS

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

Difference equations

Markov processes