Linear Algebra - Eigenvector question

AI Thread Summary
The discussion revolves around finding eigenvalues and eigenvectors for a linear operator represented by a specific matrix in a non-standard basis. The eigenvalues calculated are λ = 8 and λ = -1, but there is confusion regarding the eigenvectors due to the basis change. The correct eigenvector for λ = 8 is determined to be (-5/4, 1), which can be scaled to (-5, 4) to eliminate fractions. The final confusion arises from converting this vector to the standard basis, leading to the book's answer of (-5, -6), which is clarified through a linear combination of the basis vectors. Understanding how to express eigenvectors in the given basis is essential for solving the problem correctly.
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Homework Statement



The linear operator T on R2 has the matrix
<br /> <br /> \begin{bmatrix}4&amp;-5\\-4&amp;-3 \end{bmatrix} <br /> <br /> relative to the basis {(1,2), (0,1)}
Find the eigenvalues of T, and obtain an eigenvector corresponding to each eigenvalue.


Homework Equations





The Attempt at a Solution



So I solved the eigenvalues to be <br /> \lambda = 8, \lambda = -1<br />

I know I normally just sub in the lambda to the matrix and then solve for the null space to get the eigenvector, but how do I do it with a different basis?
 
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Is that the right matrix? I get different eigenvalues than you do.

Oh, and to answer your question, you just solve for them like you usually do. The eigenvectors you find will be represented in the given basis.
 
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Oh sorry it's supposed to be a 3 not -3..
So then I solve for it like this?

For lambda = 8
\begin{bmatrix}4&amp;-5\\-4&amp;3 \end{bmatrix} \rightarrow \begin{bmatrix}-4&amp;-5\\-4&amp; -5 \end{bmatrix} \rightarrow \begin{bmatrix}1&amp;5/4\\0&amp; 0 \end{bmatrix} <br /> <br />

Then the vector is (-5/4, 1)?

But the answer in the book is (-5, -6) :/

Ok I think I see what they did.. just multiplied it by 4 to get rid of the fraction and then wrote it wrt to the standard basis?
 
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Yup, that's what they did.
 
vela said:
Yup, that's what they did.

Hi, could you just help me with the part where they write it wrt to the basis? It sort of confuses me.

I understand we get (-5/4 1), and once clearing the fraction we get (-5, 4). But how does this jump to (-5, -6)?
 
-5(1,2) + 4(0,1)
 
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