Search results for “algebraic groups”
96 results
Eigenvectors and Eigenvalues
Theory
Eigenvectors and eigenvalues, the characteristic equation, eigenspaces, and diagonalization of matrices.
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Matrices
Theory
A matrix, its notation, and the main types: row, column, echelon, square, diagonal, identity, symmetric - defined and shown with worked examples.
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A non-Noetherian local ring with finitely generated maximal ideal
Notes
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.
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Basic Operations on Matrices
Theory
Addition, scalar multiplication, transposition and multiplication of matrices, along with subtraction, powers, and trace.
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Properties of Matrix Operations
Theory
The basic properties of addition, scalar multiplication, transposition, and multiplication of matrices, with justifications.
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Properties of Determinants
Theory
The basic properties of determinants with brief proofs, and their use in simplifying computations.
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Inverse Matrices
Theory
Nonsingular matrices, the invertibility criterion, the adjugate-matrix formula, and the Gauss–Jordan method.
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Matrix Rank
Theory
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
Theory
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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Cramer's Rule and Gaussian Elimination
Theory
Two methods for solving a SLAE — Cramer's rule via determinants and the universal Gaussian elimination method, with examples.
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Basis, Dimension, and Span
Theory
Linear dependence of vectors, basis and coordinates, dimension, subspaces, and linear span.
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Linear Operator
Theory
A linear operator and its matrix, the action x' = Ax, similar matrices, and the formula B = C⁻¹AC.
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Change of Basis
Theory
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
Theory
Quadratic forms and their matrix, definiteness, Sylvester's criterion, and canonical form.
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