Sergey

Ориентация тройки векторов, определение векторного произведения через длину, перпендикулярность и правую тройку, геометрический смысл (площадь параллелограмма), свойства с доказательствами и полный вывод формулы через определитель.

Quadratic forms and their matrix, definiteness, Sylvester's criterion, and canonical form.

The change-of-basis matrix, recomputing a vector's coordinates and an operator's matrix, and the orthonormal case.

Eigenvectors and eigenvalues, the characteristic equation, eigenspaces, and diagonalization of matrices.

A linear operator and its matrix, the action x' = Ax, similar matrices, and the formula B = C⁻¹AC.

Linear dependence of vectors, basis and coordinates, dimension, subspaces, and linear span.

The axioms of a linear (vector) space, examples and counterexamples.

Two methods for solving a SLAE — Cramer's rule via determinants and the universal Gaussian elimination method, with examples.

Linear systems in matrix form, the Kronecker–Capelli theorem, homogeneous systems, and the fundamental system of solutions.

Linear dependence and independence of the rows of a matrix, the criterion via a linear combination, and the connection to rank.

Minors and the rank of a matrix, invariance of rank under elementary transformations, and a practical method for computing it.

Nonsingular matrices, the invertibility criterion, the adjugate-matrix formula, and the Gauss–Jordan method.