Determinant via Permutations
2026TheorySergey
The direct definition of a determinant via permutations: inversions, parity, and a sum of n! terms.
0
170
Search results for “sine”
Categories
124 results
Properties of Determinants
2026TheorySergey
The basic properties of determinants with brief proofs, and their use in simplifying computations.
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260
Matrix Rank
2026TheorySergey
Minors and the rank of a matrix, invariance of rank under elementary transformations, and a practical method for computing it.
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190
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.
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560
Cramer's Rule and Gaussian Elimination
2026TheorySergey
Two methods for solving a SLAE — Cramer's rule via determinants and the universal Gaussian elimination method, with examples.
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250
Linear (Vector) Space
2026TheorySergey
The axioms of a linear (vector) space, examples and counterexamples.
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160
Eigenvectors and Eigenvalues
2026TheorySergey
Eigenvectors and eigenvalues, the characteristic equation, eigenspaces, and diagonalization of matrices.
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170
Change of Basis
2026TheorySergey
The change-of-basis matrix, recomputing a vector's coordinates and an operator's matrix, and the orthonormal case.
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220
Quadratic Forms
2026TheorySergey
Quadratic forms and their matrix, definiteness, Sylvester's criterion, and canonical form.
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160
Лекция 5. Идеалы, теорема о гомоморфизмах, евклидовы и факториальные кольца.
2026LectureAlexey Ilyin
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340
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.
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340
Векторное произведение
2026OtherSergey
Ориентация тройки векторов, определение векторного произведения через длину, перпендикулярность и правую тройку, геометрический смысл (площадь параллелограмма), свойства с доказательствами и полный вывод формулы через определитель.
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90
A curiosity: “supersmooth” varieties
2026TheorySean Cotner
A rigidity condition on schemes, strictly stronger than smoothness, introduced with examples and a look at where it fails to be geometric — no known application, just a curiosity.
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120
An example of a non-reduced Picard scheme
2026TheorySean Cotner
An example, due to Serre, of a smooth projective surface in positive characteristic whose Picard scheme fails to be reduced, worked out via the two governing dimension inequalities.
1
170
Finite flat commutative group schemes embed locally into abelian schemes
2026TheorySean Cotner
A proof of Raynaud's theorem: every finite flat commutative group scheme over a locally noetherian base embeds, Zariski-locally, as a closed subgroup of an abelian scheme.
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150
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.
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180