# Change of Basis between different size spaces

1. Feb 29, 2012

### trap101

Hi,

I'm working on an example question with the following info:

$\alpha$ = {(3,0,1) , (3,1,1), (2,1,1)} $\beta$ = {(1,1), (1,-1)} Are a set of bases. [T]$\beta\alpha$ = \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & -1 \end{bmatrix} Now they go on to say:

Let T: R3--> R3 be the transformation whose matris with respect to the basis $\alpha$ is:

[T]$\\alpha\alpha$ = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}

Now I'm trying to do some back calculations to figure out how they got that matrix, but the only thing I know would be to use a calculation such as $\beta\alpha$ [T]$\alpha\beta$

But I can't write out a vector in R2 as a linear combination of vectors in R3 right? So how would I get that

[T]$\\alpha\alpha$ = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}