Eigenvectors and Eigenvalues
Theory
Eigenvectors and eigenvalues, the characteristic equation, eigenspaces, and diagonalization of matrices.
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Search results for “algebra”
Categories
97 results
@sean_cotner
NSF Research Fellow and postdoc at the University of Bonn. Previously an NSF Research Fellow and Postdoctoral Assistant Professor at the University of Michigan, and before that a graduate student at Stanford, advised by Brian Conrad. Interested in arithmetic geometry, algebraic groups (broadly construed), and integral questions connected to the local Langlands program.
CMI 2000, The Millennium Prize Problems
Problem sheet
The Millennium Meeting, Paris, 24 May 2000, at which the seven problems were announced.
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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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The Steinberg Representation
Theory
An introduction to the Steinberg representation of a finite group of Lie type — its alternating-sum construction from parabolic inductions, worked out explicitly for SL_2.
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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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