Calculus subspace

Question

Let v=[3,3,−4,−7]. Find a basis of the subspace of R4 consisting of all vectors perpendicular to v.

Answer 1

To find a basis for the subspace of R⁴ consisting of all vectors perpendicular to v = [3, 3, -4, -7], we solve:

3x₁ + 3x₂ - 4x₃ - 7x₄ = 0

Solve for x₁:

x₁ = -x₂ + (4/3)x₃ + (7/3)x₄

Now write the general solution vector as:

x = [x₁, x₂, x₃, x₄]

= [-x₂ + (4/3)x₃ + (7/3)x₄, x₂, x₃, x₄]

Now express this as a linear combination:

x = x₂ * [-1, 1, 0, 0]

+ x₃ * [4/3, 0, 1, 0]

+ x₄ * [7/3, 0, 0, 1]

So, a basis for the subspace is:

[-1, 1, 0, 0]

[4/3, 0, 1, 0]

[7/3, 0, 0, 1]