Eigenvectors and Eigenvalues
2026TheorySergey
Eigenvectors and eigenvalues, the characteristic equation, eigenspaces, and diagonalization of matrices.
0
170
Search results for “number-theory”
87 results
Properties of Determinants
2026TheorySergey
The basic properties of determinants with brief proofs, and their use in simplifying computations.
0
260
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
Some examples of algebraic groups
2026TheorySean Cotner
Two pathological phenomena for algebraic groups over general bases — a group degenerating between the multiplicative and additive group across a DVR, and a non-affine identity component — built as centralizers in SL_n.
0
180
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
Determinant via Permutations
2026TheorySergey
The direct definition of a determinant via permutations: inversions, parity, and a sum of n! terms.
0
170
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
Basis, Dimension, and Span
2026TheorySergey
Linear dependence of vectors, basis and coordinates, dimension, subspaces, and linear span.
0
220
Determinant
2026TheorySergey
The determinant of a square matrix — its meaning, minors and cofactors, Laplace expansion, and computation methods.
0
190
Vector calculus. Green's theorem
2026Problem sheetSergey
A contour integral as a double integral: the hypotheses of Green's theorem, orientation of the contour, area as a contour integral, closing an open arc, an annulus, a singular point inside the contour.
0
260
Vector calculus. Stokes' theorem. Circulation
2026Problem sheetSergey
Circulation of a field as the flux of its curl: computing the curl, matching the orientation of the contour and the normal, circulation computed two ways.
0
310
Basic Operations on Matrices
2026TheorySergey
Addition, scalar multiplication, transposition and multiplication of matrices, along with subtraction, powers, and trace.
0
150