Vectors, matrices, transformations.
An arrow in space — magnitude and direction. Or, equivalently, an ordered list of numbers: v = (3, 2).
A vector space V is a set closed under linear combinations: take any vectors in it, scale them, add them — you stay inside.
Basis — a minimal set of vectors whose linear combinations reach every point in V. Like coordinate axes, but you choose them.
Dimension — the number of vectors in any basis. ℝ² has dim 2; the space of polynomials of degree ≤ 5 has dim 6.
If the vᵢ are linearly independent, the representation is unique — that's a basis.
A matrix is a rectangular array of numbers. We write A ∈ ℝm×n for m rows and n columns.
Matrices are not just bookkeeping — they are the linear maps.
Matrix multiplication looks bizarre until you realize: it is the composition of linear maps.
First apply B, then A. The product AB is the single matrix that does both. Hence it is not commutative — in general AB ≠ BA, just as putting on socks then shoes differs from shoes then socks.
Entry rule: (AB)ij = Σₖ aᵢₖ bₖⱼ — row of A dotted with column of B.
Shape rule: (m×k)·(k×n) = m×n. Inner dimensions must match.
A linear transformation preserves addition and scaling. Every such map on ℝⁿ is a matrix.
diag(2, 1) — pulls along an axis.
angle θ; columns are (cos θ, sin θ) and (−sin θ, cos θ).
(1, 1; 0, 1) — slants the grid.
collapses onto a subspace; loses information.
The determinant det(A) is the signed volume scaling factor of the linear map A. A unit cube of volume 1 becomes a parallelepiped of volume |det(A)|.
An eigenvector v of A is a direction that the map only scales — it does not turn.
The scalar λ is the matching eigenvalue. Eigen-pairs reveal the intrinsic axes of a transformation — the skeleton beneath the cosmetics.
Found by solving det(A − λI) = 0, the characteristic polynomial.
The fundamental equation. Given matrix A and right-hand side b, find the unknown vector x.
Existence — a solution exists iff b lies in the column space of A.
Uniqueness — the solution is unique iff the null space of A is trivial: only x = 0 maps to 0.
The rank-nullity theorem: every input dimension is either preserved (rank) or crushed (nullity). Linear algebra's conservation law.
Hard matrices become easy when factored into structured pieces. Three workhorses:
LU — A = LU. Lower-triangular times upper-triangular. Solves Ax = b in two cheap sweeps. Underlies Gaussian elimination.
QR — A = QR. Orthogonal Q times upper-triangular R. The engine of least squares.
SVD — A = UΣVᵀ. Every matrix is a rotation, then a stretch along orthogonal axes (singular values), then another rotation. The single most useful factorization in applied math: PCA, low-rank approximation, pseudo-inverse, latent semantic analysis.
Computer graphics — every rotation, translation, projection, camera transform is a 4×4 matrix. Pixels on screen are the matrix product of geometry and projection.
Machine learning — data is matrices, weights are matrices, gradients are matrices. A neural network is mostly Wx + b, repeated.
Physics — quantum mechanics: states are vectors, observables are Hermitian operators, eigenvalues are measured outcomes. Classical mechanics: rigid-body inertia tensors, normal modes, coupled oscillators.
A discipline indistinguishable from engineering, science, and economics — once you look closely enough.
Close the notebook. Open it again tomorrow. — fin.