← Module 8

Module 8: Quick Reference

Applications of Linear Algebra

Change of Basis

x = P_B [x]_B | [x]_B = P_B^{-1} x
Similar: B = P^{-1}AP (same eigenvalues, det, trace, rank)

SVD

A = U Sigma V^T (every matrix)

Singular Values

sigma_i = sqrt(eigenvalues of A^T A)

Rank

# nonzero singular values

Best rank-k approx: keep top k terms of sigma_i u_i v_i^T

PCA

Center → S = (1/(n-1)) X_c^T X_c → eigen(S) → project

PCs

Eigenvectors of S (ordered by eigenvalue)

Variance

lambda_k / sum = proportion explained

Graphics Transforms

Rotation

[cos -sin; sin cos]

Scale

[sx 0; 0 sy]

Homogeneous: (x,y) → (x,y,1) | Translation: [1 0 tx; 0 1 ty; 0 0 1]
Compose: M = T_last * ... * T_1 (rightmost first)

Perspective

(x,y,z) → (x/z, y/z)