Is my Solution for [T]B Correct?

[Nexttime35]

Homework Statement

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

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.

 

[STEMucator]

Homework Statement

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

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.

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 $[T]_B$.

 

[Nexttime35]

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

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?

Thank you for your help.

 

[STEMucator]

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?

Thank you for your help.

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.

 

[Nexttime35]

Ah, gotcha. I understand now. Thank you.