List of Symbols

A = [a_ij, matrix, 27

Ā = [ā_ij], conjugate matrix, 38

|A|, determinant, 266, 270

A*, adjoint, 379

A^H, conjugate transpose, 38

A^T, transpose, 33

A⁺, Moore–Penrose inverse, 420

A_ij, minor, 271

A(I, J), minor, 275

A(V), linear operators, 176

adj A, adjoint (classical), 273

A ~ B, row equivalence, 72

A B, congruence, 362

C, complex numbers, 11

Cⁿ, complex n-space, 13

C[a, b, continuous functions, 230

C(f), companion matrix, 306

colsp (A), column space, 120

d(u, υ), distance, 5, 243

diag(a₁₁,…, a_nn), diagonal matrix, 35

diag(A₁₁,…, A_nn), block diagonal, 40

det(A), determinant, 270

dim V, dimension, 124

{e₁,…, e_n}, usual basis, 125

E_k, projections, 386

f : A → B, mapping, 166

F(X), function space, 114

G ° F, composition, 175

Hom(V, U), homomorphisms, 176

i, j, k, 9

I_n, identity matrix, 33

Im F, image, 171

J(λ), Jordan block, 331

K, field of scalars, 112

Ker F, kernel, 171

m(t), minimal polynomial, 305

M_m,n, m × n matrices, 114

n-space, 5, 13, 229, 242

P(t), polynomials, 114

P_n(t), polynomials, 114

proj(u, υ), projection, 6, 236

proj(u, V), projection, 237

Q, rational numbers, 11

R, real numbers, 1

Rⁿ, real n-space, 2

rowsp (A), row-space, 120

S^⊥, orthogonal complement, 233

sgn σ, sign, parity, 269

span(S), linear span, 119

tr(A), trace, 33

[T]_S, matrix representation, 197

T*, adjoint, 379

T-invariant, 329

T^t, transpose, 353

||u||, norm, 5, 13, 229, 243

[u]_s, coordinate vector, 130

u · υ, dot product, 4, 13

u, υ, inner product, 228, 240

u × υ, cross product, 10

u ⊗ υ, tensor product, 398

u ^ υ, exterior product, 403

u ⊕ υ, direct sum, 129, 329

V ≊ U, isomorphism, 132, 171

V ⊗ W, tensor product, 398

V*, dual space, 351

V**, second dual space, 352

Λ^{r V, exterior product, 403}

W⁰, annihilator, 353

, complex conjugate, 12

Z(υ, T), T-cyclic subspace, 332

δ_ij, Kronecker delta, 37

Δ(t), characteristic polynomial, 296

λ, eigenvalue, 298

Σ, summation symbol, 29