Linear algebra 1000-711ALI

1. Gauss elimination method
2. Matrix algebra
3. Matrix form of systems of equations (LDU decomposition of a matrix, Gauss-Jordan method of determining the inverse matrix)
4. Linear spaces (fundamental subspaces related to matrices, linear independence of vectors,
base and dimension of a linear space, rank of a matrix)
5. Orthogonality (projection of a vector on a line and on a subspace, orthogonal complement of space,
least squares method, Gram-Schmidt orthogonalization)
6. Matrix determinant (axiomatic definition, Laplace expansion, Cramer's formulas, calculation of the volume of solids)
7. Eigenvalues and eigenvectors (remarks about complex numbers, matrix diagonalization,
exponentiation and exponential function of a matrix, spectral decomposition of a symmetric matrix, principal component analysis)
8. Singular value decomposition of matrices (positive definite and positive semidefinite matrices, SVD decomposition)