# Change of Basis Problem

1. Mar 3, 2014

### Nexttime35

1. The problem statement, all variables and given/known data

Let A = [1 0
4 2 ]
Let B be the eigenbasis {[1,4], [0,1]}.
--Find [T]B where T(x)=A(x).

3. The attempt at a solution

Would [T]B = {[1,-1], [0,2]}?

We are trying to find [T]B, the matrix representation of T with respect to B. So would my answer be correct?

Thanks.

2. Mar 3, 2014

### Zondrina

If I'm reading correctly, you want to find $[T]_B$. This amounts to finding the image of the basis vectors under $T$.

I would like to add that the eigen basis you have exhibited has some relevance as well. If you happen to know the eigenvalues you got those basis vectors with, then the diagonal matrix formed from these eigenvectors IS $[T]_B$.

Last edited: Mar 3, 2014
3. Mar 3, 2014

### Nexttime35

Yes, I want to find $[T]_B$ . I guess I am confused about how to find the basis for im(T). Could you possibly point me in the right direction?

4. Mar 3, 2014

### Zondrina

Compute $T([1 \space 4])$. Do the same for the other basis vector.

One of your vectors for $[T]_B$ was correct originally I believe.

5. Mar 3, 2014

### Nexttime35

Ah, gotcha. I understand now. Thank you.