Question

Let W be a subspace of ℝ4 spanned by the vectors:

u1 = [1; -4; 0; 1], u2 = [7; -7; -4; 1]

Find an orthogonal basis for W by performing the Gram Schmidt proces to there vectors. Find a basis for W perp (W with the upside down T).

I applied the Gram Schmidt process and found the basis for W which was {[1; -4; 0; 1], [5; 1; -4; -1].

Here's where I run into the problem. I'm trying to get the basis for W perp, but I don't really know what to do. I tried finding the basis for the transpose of W, I got x3 and x4 as the free variables, but the resulting vectors I got were wrong, any ideas?

Answer 1

I need to refamiliarize myself with the Gram-Schmidt process, but this is the question you're asking?

Given:\[u_1=\begin{bmatrix}1\\-4\\0\\1\end{bmatrix},u_2=\begin{bmatrix}7\\-7\\-4\\1\end{bmatrix}\]And that \(\left\langle u_1,u_2\right\rangle=W\) is a subspace of \(\mathbb{R}^4\), find the orthogonal bases of \(W\).