Search results for “algebraic geometry”

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

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

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

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

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.

Addition, scalar multiplication, transposition and multiplication of matrices, along with subtraction, powers, and trace.

The basic properties of addition, scalar multiplication, transposition, and multiplication of matrices, with justifications.

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.