**Linear Algebra, [1ΚΠ02 -
First Semester]**

Matrix algebra and basic properties of operations. Inverse matrices. Algorithms of computing of inverse and properties of invertible matrices. Determinants and their properties. Matrices and systems of linear equations. Rank of a matrix. Solving linear systems-Algorithms Gaus and Cramer.

Euclideom spaces . Vector spaces and subspaces. Special vector subspaces (sum, intersection, orthogonal complement). Linear combinations, spanning sets, linear dependence-independence of vectors. Basis and dimension of a vector space. Fundamental theorem for a finite dimension vector space.

Inner product on a vector space norm of a vector. Orthogonal spaces and basic theorems. Orthonormalization of a basis-Algorithm Gram-Schmidt. Linear transformation – Matrix of a linear transformation. Similar matrices. Eigenvalues and lingenvectors- Properties- Theorem Cayley- Hamilton Minimal. Polynomial. Diagonalizing of a matrix. Two criteria diagonizable matrix (linear transformation). Spectral theorem. Applications. Quadratic forms- basic criteria for symmetric matrices. Applications to quadratic forms on the problems min-max.

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