Well this question has been driving me crazy.

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

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

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

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

a) Find the change of basis matrix from B to C.

b) If the coordinate vector x relative to B is = [1 3$\displaystyle ]^t$, what is the coordinate vector x relative to C?

I dont understand if B and C can be in a subspace of $\displaystyle R^3$ 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 = $\displaystyle \begin{array}{ccc}1\\0\\1\end{array}$ $\displaystyle \begin{array}{cc}1\\1\\1\end{array}$ $\displaystyle \begin{array}{cc}1\\0\\0\end{array}$

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

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

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

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

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$\displaystyle ]_c$ , [v2$\displaystyle ]_c]$

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.