Calculus Subspace

Question

Let u = [1,0,2−4], V=[0,1,−9,−9], and let W the subspace of R4 spanned by {u,v}. Find a basis for W⊥.

Answer 1

We are given:

u = [1, 0, 2, -4]

v = [0, 1, -9, -9]

Let W be the subspace of R⁴ spanned by {u, v}. To find a basis for W⊥ (the orthogonal complement), we find all vectors x = [x₁, x₂, x₃, x₄] such that:

1. x • u = x₁ + 2x₃ - 4x₄ = 0
2. x • v = x₂ - 9x₃ - 9x₄ = 0

Solve for x₁ and x₂:

x₁ = -2x₃ + 4x₄

x₂ = 9x₃ + 9x₄

So the general solution vector x is:

x = [-2x₃ + 4x₄, 9x₃ + 9x₄, x₃, x₄]

Write this as a linear combination:

x = x₃ * [-2, 9, 1, 0]

1. x₄ * [4, 9, 0, 1]

Therefore, a basis for W⊥ is:

[-2, 9, 1, 0]

[4, 9, 0, 1]