## Finding the Change of Basis Matrix

Well this question has been driving me crazy.

Let B = {[v1,v2]} and C = {[u1,u2]}.

$v_1$ = $\begin{array}{cc}1\\0\\1\end{array}$ , $v_2$ = $\begin{array}{cc}1\\1\\1\end{array}$

$u_1$ = $\begin{array}{cc}2\\1\\2\end{array}$ , $u_2$ = $\begin{array}{cc}-1\\1\\-1\end{array}$

Then B and C are bases of the same subspace of $R^3$.

a) Find the change of basis matrix from B to C.
b) If the coordinate vector x relative to B is = [1 3 $]^t$, what is the coordinate vector x relative to C?

I dont understand if B and C can be in a subspace of $R^3$ with only two vectors.

Do I have to extend them to 3 vectors? Can I just extend it with a standard vector or do I just leave them as they are.

Well I tried to extend B and C by

B = $\begin{array}{ccc}1\\0\\1\end{array}$ $\begin{array}{cc}1\\1\\1\end{array}$ $\begin{array}{cc}1\\0\\0\end{array}$

C = $\begin{array}{cc}2\\1\\2\end{array}$ $\begin{array}{cc}-1\\1\\-1\end{array}$ $\begin{array}{cc}1\\0\\0\end{array}$

Then using the augmented matrix [C B] I used gaussian elimination to get the change of basis matrix D.

D = $\begin{array}{cc}\frac{1}{3}\\\frac{-1}{3}\\0\end{array}$ $\begin{array}{cc}\frac{2}{3}\\\frac{1}{3}\\0\end{a rray}$ $\begin{array}{cc}0\\0\\1\end{array}$

I also checked this by working out the inverse of C, so the inverse of C times B = D and I got the same answer to D.

Then I got stuck on question b) because as I understand to work out the coordinate vector x relative to C:

[x $]_c$ = D[x $]_b$

but I cant multiply D (3x3 matrix) by a 2x1 matrix, so Im not sure do I have to extend the coordinate vectors aswell?

I also attempted it another way by trying to find D = [[v1 $]_c$ , [v2 $]_c]$

Where:
2 -1 1 1
1 1 0 1
2 -1 1 1
-----------
-3 1 -1
-3 1 -1

Then I got -3[Row 2] = [1 -1] and back subtituted into the first piviot row and the result was a 3x2 matrix where I could multiply by the coordinate vector relative to B inorder to work out b).

I dont know where I stand as to how to completely answer this question, your help would be much appreciated. I think I may have missed some important step. or thereom. Help.