Properties of Determinants
2026TheorySergey
The basic properties of determinants with brief proofs, and their use in simplifying computations.
0
260
Search results for “algebraic groups”
50 results
Inverse Matrices
2026TheorySergey
Nonsingular matrices, the invertibility criterion, the adjugate-matrix formula, and the Gauss–Jordan method.
0
120
Matrix Rank
2026TheorySergey
Minors and the rank of a matrix, invariance of rank under elementary transformations, and a practical method for computing it.
0
190
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.
0
560
Cramer's Rule and Gaussian Elimination
2026TheorySergey
Two methods for solving a SLAE — Cramer's rule via determinants and the universal Gaussian elimination method, with examples.
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250
Linear (Vector) Space
2026TheorySergey
The axioms of a linear (vector) space, examples and counterexamples.
0
160
Basis, Dimension, and Span
2026TheorySergey
Linear dependence of vectors, basis and coordinates, dimension, subspaces, and linear span.
0
220
Linear Operator
2026TheorySergey
A linear operator and its matrix, the action x' = Ax, similar matrices, and the formula B = C⁻¹AC.
0
150
Eigenvectors and Eigenvalues
2026TheorySergey
Eigenvectors and eigenvalues, the characteristic equation, eigenspaces, and diagonalization of matrices.
0
170
Change of Basis
2026TheorySergey
The change-of-basis matrix, recomputing a vector's coordinates and an operator's matrix, and the orthonormal case.
0
220
Quadratic Forms
2026TheorySergey
Quadratic forms and their matrix, definiteness, Sylvester's criterion, and canonical form.
0
160
A non-Noetherian local ring with finitely generated maximal ideal
2026NotesBogdan Zavyalov
A complete local ring with a finitely generated maximal ideal is Noetherian. Drop completeness and it fails: an ultraproduct construction gives a non-Noetherian local ring whose maximal ideal is principal.
0
340