Inner product spaces:

- norm induced by an inner product

- Pythagoras theorem, Cauchy-Schwarz inequality, triangle inequality

- orthogonal and orthonormal system of vectors, Fourier coefficients, Gram-Schmidt orthogonalization

- orthogonal complement, orthogonal projection

- the least squares method

- orthogonal matrices

Determinants:

- basic properties

- Laplace expansion of a determinant, Cramer's rule

- adjugate matrix

- geometric interpretation of determinants

Eigenvalues and eigenvectors:

- basic properties, characteristic polynomial

- Cayley-Hamilton theorem

- similarity and diagonalization of matrices, spectral decomposition, Jordan normal form

- symmetric matrices and their spectral decomposition

- (optionally) companion matrix, estimation and computation of eigenvalues: Gershgorin discs and power method

Positive semidefinite and positive definite matrices:

- characterization and properties

- methods: recurrence formula, Cholesky decomposition, Gaussian elimination, Sylvester's criterion

- relation to inner products

Bilinear and quadratic forms:

- forms and their matrices, change of a basis

- Sylvester's law of inertia, diagonalization, polar basis

Topics on expansion (optionally):

- eigenvalues of nonnegative matrices

- matrix decompositions: Householder transformation, QR, SVD, Moore-Penrose pseudoinverse of a matrix

Continuation of MAI057 - special matrices, determinants, eigenvalues, examples of applications of linear algebra.