← Module 7

Module 7: Quick Reference

Inner Product Spaces -- Linear Algebra

Dot Product & Length

u · v = u^T v = u1v1 + u2v2 + ... + unvn
||u|| = sqrt(u · u) | u-hat = u/||u|| | dist(u,v) = ||u-v||
Orthogonal: u · v = 0 | Cauchy-Schwarz: |u · v| ≤ ||u|| ||v||

Projections

proj_u(y) = [(y · u)/(u · u)] u
proj_W(y) = sum of [(y · ui)/(ui · ui)] ui (orthogonal basis required)
y = proj_W(y) + z, z in W-perp | dim(W) + dim(W-perp) = n

Gram-Schmidt

v1 = x1 | v2 = x2 - proj_{v1}(x2) | v3 = x3 - proj_{v1}(x3) - proj_{v2}(x3)

Output

Orthogonal basis for same subspace

QR Factorization

A = QR, Q orthonormal, R upper triangular

Least Squares

A^T A x-hat = A^T b

What it minimizes

||b - Ax|| (sum of squared residuals)

Best-fit line

A = [1 t1; ...; 1 tn], solve for (c0, c1)