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Change of Basis Problem

  • Thread starter Nexttime35
  • Start date
  • #1
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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.
 

Answers and Replies

  • #2
STEMucator
Homework Helper
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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##.
 
Last edited:
  • #3
46
1
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.
 
  • #4
STEMucator
Homework Helper
2,075
140
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.
 
  • #5
46
1
Ah, gotcha. I understand now. Thank you.
 

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