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The transformation $R^2 -> R^2$ is represented by the matrix $\pmatrix{1&2\\1&0}$.

Find the eigenvalues and eigenvectors and decide the matrix representation of T with respect to a basis of eigenvectors.

So, I get that the eigenvalues are: $\lambda_1 = 2 $ and $ \lambda_2 = -1 $

This gives me the eigenvectors: $ \pmatrix{2\\1}$ and $ \pmatrix{-1\\1}$. However, I don't really grasp what to do when I want to represent the matrix transformation with respect to the eigenvectors. I've tried virtually everything I could think of but everything is wrong. What's the correct way to do this?

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Note that the representation with respect to the basis of eigenvectors $v_1=\pmatrix{2\\1}$ and $v_2=\pmatrix{-1\\1}$ is the diagonal matrix with the eingenvalues on the diagonal, that is

$$\pmatrix{2&0\\0&-1}$$

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  • $\begingroup$ What is the theory behind that? Don't I want to do $ a \pmatrix{2\\1} + b \pmatrix{-1\\1} = \pmatrix{1\\1} $ to get the first column of the matrix in the linear transformation? $\endgroup$ – gbgult Apr 1 '18 at 20:17
  • $\begingroup$ @gbgult I add this part ti clarify! $\endgroup$ – gimusi Apr 1 '18 at 20:18
  • $\begingroup$ Oh, I think I get it now. I basically use that $Ax = \lambda x$, am I correct? Since x is the basis vector it's then pretty obvious that $\lambda$ is the coordinate with respect to that basis. $\endgroup$ – gbgult Apr 1 '18 at 20:21
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    $\begingroup$ @gbgult Yes exactly! $\endgroup$ – gimusi Apr 1 '18 at 20:25

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