Linear algebra - change of basis matrix

[zeion]

Homework Statement

Let A = {(1, 1), (2,0)} and B = {(0, 2), (2, 1)} in R2.
a) Find A (u with respect to A) if B = [3, -2].

Homework Equations

The Attempt at a Solution

I tried to find AB (transition matrix from B to A), then apply to B, but couldn't represent (2, 1) with respect to A?

So I found u by BB = u,
then u = (4, -4).

Now represent u with respect to A:
A = (-4, 4)

Is this correct?

 

[boaz]

Yes, that's how it's done. You don't need to bother too much.
We know that _B=[3,-2]^t => u=3*(0,2)-2*(2,1)=(-4,4).
Now, in order to find _A we need to find a,b in R such that :
a*(1, 1)+b*(2,0)=(-4,4) => a+2b=-4 and a=4 => a=4 and b=-4.
Therefore : _A = [4,-4]^t.
DONE! :)