I am looking at a homework problem I turned in and the posted solution and I still am having trouble making sense of the question.
Given two vectors:
$v_1$ = $\begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}$ $\>$$\>$ $v_2$ = $\begin{bmatrix}0 \\ -1 \\ 1\end{bmatrix}$
a) use the Gram-Schmidt procedure to perform orthonormalization on the basis {$v_1$,$v_2$}.
I was able to do this with the result of:
$q_1$ = $\frac{1}{\sqrt{2}}$ $\begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}$ $\>$$\>$ $q_2$ = $\frac{1}{\sqrt{1.5}}$$\begin{bmatrix}\frac{-1}{2} \\ -1 \\ \frac{1}{2}\end{bmatrix}$
b) Express the new basis in terms of the old basis
However, here I get turned around. How do I generate the change of basis matrix Q that will transform from $v_i$ to $q_i$ and back?
The second part of the question asks you to find scalars $a_{ij}$ such that $q_1=a_{11}v_1+a_{12}v_2$ and $q_2=a_{21}v_1+a_{22}v_2$. The Gram-Schmidt process that you just performed gives you a way to find these coefficients. You just have to save some information from the steps you took.
You no doubt began by setting $q_1={1\over\|v_1\|}v_1$, so obviously $a_{11}={1\over\|v_1\|}$ and $a_{12}=0$. When computing $q_2$, you would first have computed $$w_2=v_2-(v_2\cdot q_1)q_1=v_2-{v_2\cdot v_1\over v_1\cdot v_1}v_1$$ and then normalized the result. It should be clear, then, that the remaining two coefficients $a_{21}$ and $a_{22}$ that you’re looking for are the coefficients of $v_1$ and $v_2$ in the right-hand side above, divided by $\|w_2\|$.