Sergey

@aethereal

Math and CS

1 follower39 works

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Eigenvectors and Eigenvalues

Jul 24, 2026TheorySergey

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

170

Linear Operator

Jul 24, 2026TheorySergey

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

150

Basis, Dimension, and Span

Jul 24, 2026TheorySergey

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

220

Linear (Vector) Space

Jul 24, 2026TheorySergey

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

160

Cramer's Rule and Gaussian Elimination

Jul 24, 2026TheorySergey

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

250

Systems of Linear Equations

Jul 24, 2026TheorySergey

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

210

Linear Dependence and Independence of Rows

Jul 24, 2026TheorySergey

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

560

Matrix Rank

Jul 24, 2026TheorySergey

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

190

Inverse Matrices

Jul 24, 2026TheorySergey

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

120

Properties of Determinants

Jul 24, 2026TheorySergey

The basic properties of determinants with brief proofs, and their use in simplifying computations.

260

Determinant via Permutations

Jul 24, 2026TheorySergey

The direct definition of a determinant via permutations: inversions, parity, and a sum of n! terms.

170

Determinant

Jul 24, 2026TheorySergey

The determinant of a square matrix — its meaning, minors and cofactors, Laplace expansion, and computation methods.

190