Linear Algebra glossary
Clear, one-line definitions of the Linear Algebra terms used across the OgbonLab textbooks. Each entry links to the interactive sections where the idea is taught.
18 terms
- basis
- A linearly independent spanning set of a vector space V; every vector in V is a unique linear combination of basis vectors.
- characteristic equation
- For a square matrix A, the polynomial equation det(A − λI) = 0 whose roots are the eigenvalues of A.
- characteristic-equation
- For a square matrix A, the polynomial equation det(A − λI) = 0 whose roots are the eigenvalues of A.
- cofactor
- The signed minor Cᵢⱼ = (−1)^(i+j) Mᵢⱼ of a matrix entry; cofactors are the building blocks of determinant expansions.
- determinant
- A scalar computed from a square matrix; nonzero exactly when the matrix is invertible.
- See: The 2x2 Determinant, The 3x3 Determinant
- dimension
- The number of vectors in any basis of a vector space V; all bases of V have the same size, written dim(V).
- See: Basis, Span, and Dimension
- eigenvalue
- A scalar λ for which there exists a nonzero vector v with Av = λv; λ measures how A stretches v along its eigenvector direction.
- See: Eigenvalues, Eigenvectors, and Diagonalization
- eigenvector
- A nonzero vector v satisfying Av = λv for some scalar λ; v keeps its direction (or reverses) under the linear map A.
- See: Eigenvalues, Eigenvectors, and Diagonalization
- kernel
- For a linear map T: V → W, the subspace ker(T) = {v ∈ V : T(v) = 0}; T is injective iff ker(T) = {0}.
- See: Kernel density estimation
- linear equation
- An equation in which each variable appears to the first power and is not multiplied by another variable.
- linear-combination
- A sum c₁v₁ + c₂v₂ + ... + cₖvₖ of vectors weighted by scalars cᵢ; the basic building block of linear algebra.
- linear-transformation
- A map T: V → W between vector spaces that preserves addition and scalar multiplication: T(u+v) = T(u)+T(v) and T(cv) = cT(v).
- linearly-independent
- Vectors v₁,...,vₖ are linearly independent when c₁v₁ + ... + cₖvₖ = 0 forces every cᵢ = 0; none is redundant.
- matrix
- A rectangular array of numbers arranged in rows and columns; encodes a linear transformation.
- See: The Matrix-Density / U Plot, Matrix Operations and Algebra
- similar-matrices
- Square matrices A and B are similar when B = P⁻¹AP for some invertible P; similar matrices share eigenvalues, determinant, and trace.
- span
- The set span(S) of all linear combinations of vectors in S; the smallest subspace containing S.
- system of equations
- A collection of equations sharing the same variables; a solution satisfies every equation simultaneously.
- vector-space
- A set V with addition and scalar multiplication over a field, satisfying associativity, commutativity, distributivity, and identity/inverse axioms.