A matrix, its notation, and the main types: row, column, echelon, square, diagonal, identity, symmetric - defined and shown with worked examples.
How to bring up a twenty-year-old Unix TCAD package under emulation on an ARM Mac, drive it from Jupyter, and parse its data formats in Python
A pilot study mapping ion-induced charge collection against bandgap and carrier mobility, and what it revealed about the cost of getting the time integration wrong
Every finite flat commutative group scheme over a noetherian local ring is the torsion component of the Picard scheme of a smooth projective scheme with 3-dimensional fibers, built as a quotient of a complete intersection. An application: Hodge numbers that jump in a smooth projective family.
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.
Addition, scalar multiplication, transposition and multiplication of matrices, along with subtraction, powers, and trace.
The basic properties of addition, scalar multiplication, transposition, and multiplication of matrices, with justifications.
The determinant of a square matrix — its meaning, minors and cofactors, Laplace expansion, and computation methods.
The direct definition of a determinant via permutations: inversions, parity, and a sum of n! terms.
The basic properties of determinants with brief proofs, and their use in simplifying computations.
Nonsingular matrices, the invertibility criterion, the adjugate-matrix formula, and the Gauss–Jordan method.
Minors and the rank of a matrix, invariance of rank under elementary transformations, and a practical method for computing it.
Linear dependence and independence of the rows of a matrix, the criterion via a linear combination, and the connection to rank.
Linear systems in matrix form, the Kronecker–Capelli theorem, homogeneous systems, and the fundamental system of solutions.
Two methods for solving a SLAE — Cramer's rule via determinants and the universal Gaussian elimination method, with examples.
The axioms of a linear (vector) space, examples and counterexamples.
Linear dependence of vectors, basis and coordinates, dimension, subspaces, and linear span.
A linear operator and its matrix, the action x' = Ax, similar matrices, and the formula B = C⁻¹AC.
Eigenvectors and eigenvalues, the characteristic equation, eigenspaces, and diagonalization of matrices.
The change-of-basis matrix, recomputing a vector's coordinates and an operator's matrix, and the orthonormal case.