← Module 4

Module 4: Quick Reference

Vector Spaces

Vectors

Dot Product

u.v = sum u_i*v_i. Result: scalar.

Norm

||v|| = sqrt(v.v). Unit: v/||v||.

Orthogonal: u.v = 0

Subspace Test

(1) Contains 0. (2) Closed under +. (3) Closed under scalar mult.

Independence

Test

Row reduce, check for free vars. No free vars = independent.

Key Facts

  • More than n vecs in R^n: dependent
  • Zero vector in set: dependent

Basis & Dimension

Basis = independent + spanning. dim = #vectors in basis.

Col(A) basis

Pivot columns of ORIGINAL A.

Nul(A) basis

Direction vectors from Ax=0 parametric form.

Rank Theorem

rank + nullity = n (# columns)