Obtain an eigenvector corresponding to each eigenvalue

[jesuslovesu]

Homework Statement

The linear operator T on R^2 has the matrix [4 -5; -4 3]
relative to the basis { (1,2), (0,1) }

Find the eigenvalues of T.
Obtain an eigenvector corresponding to each eigenvalue.

Homework Equations

The Attempt at a Solution

I was able to find the eigenvalues (8 and -1) easily enough; however, I have not been able to find the eigenvectors. (I have a feeling it's due to a nonstandard basis being given.)

AX = 8X
AX = -1X

In the case of the first equation I get 4a - 5b = 8a; -4a + 3b = 8b so I would think an eigenvector could be (-5, 4) however my book says it should be (-5,-6). I believe this is because of the basis, but I really don't know what to do with it.

 

[mjsd]

are you sure it is (-5,-6)? or did u write down the basis correctly? {(1,2),(0,1)}

EDIT: ok i think it is correct, my mistakes

 

[jesuslovesu]

Yep, according to my book the answer is (-5,-6) and the basis is correct, is the book wrong?

 

[mjsd]

is the eigenvector corresponding to evalue 1 is (1,3) according to book?

 

[jesuslovesu]

yep.. I can see that (-5,4) -> -5(1,2) + 4(0,1) = (-5,-6)
is (-5,4) a coordinate matrix?

 

[mjsd]

yes... but I can't pinpoint the cause of the problem in your method just yet...

 

[mjsd]

by the way, i believe the answers in the book is written in terms of the standard basis {(1,0), (0,1)}

 

[mjsd]

ok, previously I made two mistakes that confused myself
note: -5 (1,2) +4 (0,1) = (-5,-6)
and 1 (1,2) +1 (0,1) = (1,3)

 

[jesuslovesu]

alright i think i can see it now, thanks