Properties of Determinants
2026TheorySergey
The basic properties of determinants with brief proofs, and their use in simplifying computations.
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Inverse Matrices
2026TheorySergey
Nonsingular matrices, the invertibility criterion, the adjugate-matrix formula, and the Gauss–Jordan method.
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Matrix Rank
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Minors and the rank of a matrix, invariance of rank under elementary transformations, and a practical method for computing it.
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Linear Dependence and Independence of Rows
2026TheorySergey
Linear dependence and independence of the rows of a matrix, the criterion via a linear combination, and the connection to rank.
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Systems of Linear Equations
2026TheorySergey
Linear systems in matrix form, the Kronecker–Capelli theorem, homogeneous systems, and the fundamental system of solutions.
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Cramer's Rule and Gaussian Elimination
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Two methods for solving a SLAE — Cramer's rule via determinants and the universal Gaussian elimination method, with examples.
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Linear (Vector) Space
2026TheorySergey
The axioms of a linear (vector) space, examples and counterexamples.
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Basis, Dimension, and Span
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Linear dependence of vectors, basis and coordinates, dimension, subspaces, and linear span.
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Linear Operator
2026TheorySergey
A linear operator and its matrix, the action x' = Ax, similar matrices, and the formula B = C⁻¹AC.
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Eigenvectors and Eigenvalues
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Eigenvectors and eigenvalues, the characteristic equation, eigenspaces, and diagonalization of matrices.
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Change of Basis
2026TheorySergey
The change-of-basis matrix, recomputing a vector's coordinates and an operator's matrix, and the orthonormal case.
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Quadratic Forms
2026TheorySergey
Quadratic forms and their matrix, definiteness, Sylvester's criterion, and canonical form.
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