## Topic outline

### Determinants

Definition, properties with respect to matrix operations - transposition, elementary row operations, matrix product, inverse

Linearity of the determinant

Determinant calculation - by Gauss elimination and by Laplace expansion

Adjoint matrix

Cramer's rule

Geometric meaning of the determinant

Application - number of spanning trees of a graph

### Polynomials

Polynomial operations - addition, subtraction, products, division with a remainder

Fermat's little theorem

Roots, the fundamental theorem of algebra (without a proof)

Decomposition into monomials

Representation of polynomials

Vandermonde matrix and its regularity

Lagrange interpolation

Applications - secret sharing, fast integer multiplication

### Eigenvalues and eigenvectors

Eigenvalues and eigenvectors of linear maps and square matrices

Properties of eigenvectors (subspaces, linear independence)

Characteristic polynomial and its coefficients

Calculation of eigenvalues and eigenvectors

Application in systems of differential equations

Cayley-Hamilton theorem

### Diagonalization

Similar matrices

Diagonalization - definition, existence via eigenvectors

Jordan normal form (without a proof)

Diagonalization of symmetric and Hermitian matrices

### Inner spaces

Inner product, norm

Cauchy-Schwarz inequality

Orthogonality, orthonormal bases

Orthogonal projection, Gramm-Schmidt orthonormalization

Isometry

Orthogonal complement

### Positive definite matrices

Gram matrix

Positive definite matrices

Cholesky factorization

Other characterizations

### Bilinear and quadratic forms

Bilinear and quadratic forms

Matrices of forms

Diagonalization of quadratic forms

Sylvester's law of inertia

### Applications

Number of spanning trees - see the section of determinants

Number of even subgraphs

Maximum number of lines spanning the same angle