Linear Algebra

Vectors, matrices, determinants, eigenvalues, and the structural identities of linear maps.

Vectors & Spaces

Vectors & Dot Products — Lengths come from v · v, and orthogonality is exactly v · w = 0.
Linear Independence — Independent vectors contribute genuinely new directions; dependent ones hide a linear relation.
Span & Basis — Span tells you what vectors you can build; a basis builds everything with no redundancy.
Linear Transformations — A linear map preserves addition and scaling; matrices are its coordinate form, with image and kernel as the key outputs.
Change of Basis — x = P[x]_B, [x]_B = P^-1x, and A' = P^-1AP is the same map in a new basis.

Matrix Terminology — Identity, diagonal, symmetric, orthogonal, transpose, trace — the vocabulary.
Matrix Multiplication — Entry (i,j) of AB is row i of A dotted with column j of B; in general AB ≠ BA.
Row Reduction — Gaussian elimination produces REF, preserving rank, row space, and null space.
Matrix Inverse — The 2× 2 inverse formula, invertibility via A ≠ 0, and solving Ax = b.
Determinant Basics — bmatrix a & b c & d bmatrix = ad - bc.
Determinant Properties — Row operations, the product rule (AB) = (A) (B), and as a signed volume factor.
Laplace Expansion — (A) = _j (-1)^i+j a_ij M_ij — expand along any row or column.
Cramer's Rule — x_i = (A_i)/ (A) — solve Ax = b with three determinants.

Eigenvalues Basics — Solve λ^2 - ( tr)λ + = 0 — trace and determinant do the work.
Characteristic Polynomial — p_A(λ) = (λ I - A) — its roots are the eigenvalues.
Eigenvectors & Eigenspaces — Eigenvectors solve (A - λ I)v = 0; the eigenspace is (A - λ I).
Diagonalization — A = PDP^-1 — eigenvalues in D, eigenvectors in P, and A^k = PD^kP^-1.
Spectral Theorem — Symmetric matrices: real eigenvalues, orthogonal eigenvectors, A = Q Q^T.
Matrix Rank — Rank = number of independent rows = number of independent columns.
Null Space & Nullity — null(A) = x : Ax = 0. Nullity = its dimension.
Rank-Nullity Theorem — rank(A) + nullity(A) = n — every column counts exactly once.

Operator Norm — |A|_ op = _|x|=1 |Ax| = _ (A).
Cauchy-Binet Identity — (AB) = _S (A_S) (B_S) — determinants of products via minors.
Sylvester's Determinant Identity — (I_m + AB) = (I_n + BA) — swap to the smaller side.
Hadamard's Inequality — | (A)| ≤ _i |r_i|_2 — equality if and only if rows are orthogonal.
Weyl's Inequality — Hermitian eigenvalues are stable: | _k(A+B) - _k(A)| ≤ |B|_ op.
Sherman-Morrison — (A + uv^ )^-1 = A^-1 - A^-1 u v^ A^-11 + v^ A^-1 u — rank-1 update, rank-1 correction.
Newton's Identities — Recursively translate between power sums p_k and elementary symmetric polynomials e_k.

Orthogonality & Orthonormal Sets — Dot product, norm, unit vectors, and orthonormal sets.
Inner Product Spaces — Inner product axioms, the induced norm, and Cauchy–Schwarz.
Orthogonal Projections — Closest point on a subspace, and the projection matrix.
Gram–Schmidt Process — Turn any basis into an orthonormal one by subtracting projections.
Least Squares — Best fit when there's no exact solution: the normal equations.
QR Factorization — Gram–Schmidt as a matrix product: A = QR.
Singular Value Decomposition (SVD) — Factor any matrix as A = U V^ via the eigenvalues of A^ A.
Positive Definite Matrices — Certify x^T A x > 0 via eigenvalues, pivots, and minors.
Orthogonal & Unitary Matrices — Orthonormal columns, Q^TQ = I, and preserving every length.
Jordan Normal Form — The canonical form for operators that cannot be diagonalized.