Glossary

The transposed matrix of cofactors; A^{-1}=\frac{1}{\det A}\operatorname{adj}(A).

How many times an eigenvalue repeats as a root of the characteristic polynomial.

A system's coefficients and constants compressed into one grid.

A variable corresponding to a pivot column; determined once free variables are set.

A set that both spans a space and is linearly independent.

The matrix P converting \mathcal B-coordinates to standard coordinates.

\det(A-\lambda I) = 0 — solved for \lambda, its roots are A's eigenvalues.

p_A(\lambda)=\det(A-\lambda I), a degree-n polynomial whose roots are A's eigenvalues.

The m\times n array of a system's coefficients, without the constants column.

A minor with a checkerboard sign attached: C_{ij}=(-1)^{i+j}M_{ij}.

Computing a determinant by summing entries times signed minors along any row or column.

\operatorname{Col}(A) — the span of a matrix's columns.

[\mathbf x]_{\mathcal B} — the unique list of coefficients expressing \mathbf x in a given basis.

Solving for one variable of A\mathbf x=\mathbf b at a time, as a ratio of two determinants.

A vector, defined only in \mathbb{R}^3, perpendicular to both inputs.

An eigenvalue (or matrix) whose geometric multiplicity falls short of its algebraic multiplicity.

A single number, \det A, reporting a matrix's signed scaling factor.

\det A \ne 0 exactly when a square matrix A is invertible.

A square matrix whose only nonzero entries (if any) lie on the main diagonal.

A = PDP^{-1} for some invertible P and diagonal D; equivalent to having n independent eigenvectors.

\dim V — the number of vectors in any basis of V.

The length of the vector connecting two points, d(P,Q) = \lVert P - Q\rVert.

\mathbf u \cdot \mathbf v = u_1v_1 + \cdots + u_nv_n — a single number, not a vector.

\operatorname{Nul}(A-\lambda I) — every eigenvector for a given \lambda, plus \mathbf 0.

\lambda in A\mathbf v = \lambda\mathbf v — the stretch factor belonging to an eigenvector.

A nonzero vector A only stretches or shrinks, never turns off its own line.

The identity matrix after one elementary row operation; multiplying by it performs that operation.

Swap, scale, or replace a row — the three legal, reversible moves on a matrix.

A variable corresponding to a free (non-pivot) column; a genuine degree of freedom.

Ordinary elimination plus a backward pass, reaching RREF directly.

The column-by-column procedure that reduces any matrix to row echelon form.

The dimension of an eigenvalue's eigenspace — how many independent eigenvectors it actually supplies.

Converting any basis into an orthogonal basis of the same subspace.

A\mathbf x = \mathbf 0 — a system whose right-hand side is entirely zero.

I_n — 1s on the diagonal, 0s elsewhere; the matrix analogue of 1.

\operatorname{im}(T) — every output a linear transformation can produce.

Any operation on two vectors obeying symmetry, additivity, homogeneity, and positive definiteness.

A^{-1} — the matrix with AA^{-1}=A^{-1}A=I.

A list of conditions on a square matrix that all hold together, or all fail together.

A linear map T with an S that perfectly undoes it, S(T(\mathbf x))=\mathbf x.

A bijective linear map; two spaces linked this way are V \cong W.

\ker T — every input a linear transformation sends to the zero vector.

\hat{\mathbf x} — the \mathbf x minimizing \lVert A\mathbf x - \mathbf b\rVert^2.

\operatorname{Nul}(A^{\mathsf T}) — vectors \mathbf y with \mathbf y^{\mathsf T}A=\mathbf 0.

A sum of scaled vectors, c_1\mathbf v_1 + \cdots + c_k\mathbf v_k.

a_1x_1+\cdots+a_nx_n=b — every variable alone, never multiplied or raised to a power.

No vector in the set can be built from the others; only the trivial combination gives \mathbf 0.

A map preserving addition and scalar multiplication: T(a\mathbf u+b\mathbf v)=aT(\mathbf u)+bT(\mathbf v).

A = LU — elimination's multipliers (L) times the resulting triangular matrix (U).

A system moving between states where next-state odds depend only on the current state.

A rectangular array of numbers, with shape m\times n.

A\mathbf x = \mathbf b — a whole linear system collapsed into one line.

(AB)_{ij} is row i of A dotted with column j of B.

The determinant of the smaller matrix left after deleting one row and one column.

A vector's length, \lVert\mathbf v\rVert = \sqrt{\mathbf v\cdot\mathbf v}.

A^{\mathsf T}A\hat{\mathbf x}=A^{\mathsf T}\mathbf b — the algebraic form of least squares.

\operatorname{Nul}(A) — every \mathbf x with A\mathbf x = \mathbf 0.

\operatorname{nullity}(A) = \dim\operatorname{Nul}(A), the number of free variables.

Distinct inputs always produce distinct outputs; injective.

Every vector in the codomain is hit by at least one input; surjective.

Perpendicular — two vectors with dot product exactly 0.

W^\perp — every vector orthogonal to all of a subspace W at once.

Every \mathbf y splits uniquely into \hat{\mathbf y}\in W plus a piece orthogonal to W.

Diagonalizing a symmetric matrix using an orthonormal eigenbasis, so P^{-1}=P^{\mathsf T}.

A square matrix whose columns form an orthonormal set; Q^{-1}=Q^{\mathsf T}.

\operatorname{proj}_W(\mathbf v) — summing single-vector projections onto an orthogonal basis of W.

A collection of vectors, every distinct pair of which is orthogonal.

An orthogonal set whose vectors also all have length 1.

A solution set written as \mathbf p + t\mathbf d, a base point plus a direction times a parameter.

The identity matrix with its rows reordered; used to record row swaps.

A row's leading (first nonzero) entry, reading left to right.

A column of row echelon form that contains a leading entry.

The shadow of \mathbf u cast onto the line through \mathbf v.

A = QR — orthonormal columns (Q) times an upper-triangular record of Gram-Schmidt (R).

The fraction of a dataset's total variation a least-squares fit explains.

\operatorname{rank}(A) — the shared dimension of the row space and column space.

Row echelon form with every pivot equal to 1 and cleared above as well as below.

\mathbf r = \mathbf b - A\mathbf x — the gap between actual and predicted output.

A staircase shape: leading entries march right going down, zero rows at the bottom.

Two matrices reachable from each other by elementary row operations.

\operatorname{Row}(A) — the span of a matrix's rows.

Rescaling every entry of a vector by the same number.

Square matrices A,B with B = P^{-1}AP for some invertible P.

A square matrix with no inverse.

Every tuple that satisfies an equation or system — a point, line, plane, or more.

The set of every linear combination of a given collection of vectors.

The matrix A with T(\mathbf x) = A\mathbf x, columns T(\mathbf e_i).

A subset of a vector space that is itself a complete vector space.

A square matrix equal to its own transpose, A = A^{\mathsf T}.

\operatorname{tr}(A) — the sum of a matrix's diagonal entries; also equal to the sum of its eigenvalues.

P — a column-stochastic matrix whose (i,j) entry is the probability of moving from state j to state i.

A^{\mathsf T} — rows of A rewritten as columns.

Upper triangular: zero below the diagonal. Lower triangular: zero above.

\mathbf x = \mathbf 0 — the solution every homogeneous system always has.

A vector of length exactly 1, pure direction with no length attached.

An ordered list of numbers, viewable as an arrow, a list, or a point.

Adding two vectors component by component.

A set with addition and scalar multiplication obeying ten algebraic axioms.

The vector \mathbf{0} whose every entry is 0; the additive identity.