Search results for “algebraic-geometry”

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

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

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

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

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

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

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

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

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.

A matrix, its notation, and the main types: row, column, echelon, square, diagonal, identity, symmetric - defined and shown with worked examples.