Linear algebra and analytic geometry. Vector algebra
2026Problem sheetSergey
Vectors in coordinates: dividing a segment, midpoints of sides, the angle between vectors, the cross product and the scalar triple product.
0
180
Search results for “linear algebra”
Categories
25 results
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
Systems of Linear Equations
2026TheorySergey
Linear systems in matrix form, the Kronecker–Capelli theorem, homogeneous systems, and the fundamental system of solutions.
0
210
Linear (Vector) Space
2026TheorySergey
The axioms of a linear (vector) space, examples and counterexamples.
0
160
Linear Operator
2026TheorySergey
A linear operator and its matrix, the action x' = Ax, similar matrices, and the formula B = C⁻¹AC.
0
150
Basis, Dimension, and Span
2026TheorySergey
Linear dependence of vectors, basis and coordinates, dimension, subspaces, and linear span.
0
220
Basic Operations on Matrices
2026TheorySergey
Addition, scalar multiplication, transposition and multiplication of matrices, along with subtraction, powers, and trace.
0
150
Properties of Matrix Operations
2026TheorySergey
The basic properties of addition, scalar multiplication, transposition, and multiplication of matrices, with justifications.
0
180
Determinant
2026TheorySergey
The determinant of a square matrix — its meaning, minors and cofactors, Laplace expansion, and computation methods.
0
190
Determinant via Permutations
2026TheorySergey
The direct definition of a determinant via permutations: inversions, parity, and a sum of n! terms.
0
170
Properties of Determinants
2026TheorySergey
The basic properties of determinants with brief proofs, and their use in simplifying computations.
0
260
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
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.
0
250
Eigenvectors and Eigenvalues
2026TheorySergey
Eigenvectors and eigenvalues, the characteristic equation, eigenspaces, and diagonalization of matrices.
0
170
Change of Basis
2026TheorySergey
The change-of-basis matrix, recomputing a vector's coordinates and an operator's matrix, and the orthonormal case.
0
220
Quadratic Forms
2026TheorySergey
Quadratic forms and their matrix, definiteness, Sylvester's criterion, and canonical form.
0
160
CMI 2000, The Millennium Prize Problems
2026Problem sheet
The Millennium Meeting, Paris, 24 May 2000, at which the seven problems were announced.
0
360