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The matrix $A$ has the eigenvalues $\lambda_1 = 1$ and $\lambda_2 = 4$ as well as the eigenvectors $v_1 = (2, 3)$ to $\lambda_1$ and $v_2 = (1, 2)$ to $\lambda_2$. From this information determine the matrix $A$ with respect to the canonical basis $( e_1, e_2)$.

I couldn't find a clue about the question on the internet. I will be glad if you can help me with the solution.

Nerrit
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A O
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  • I think that $e_1=2v_1-3v_2$ and $e_2=2v_1-v_1$. Now you can use linearity to determine $Ae_1$ and $Ae_2$. – Filippo Jan 03 '23 at 10:54
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    No trick needed. Name $a,b,c,d$ the four entries of your matrix and translate into equations the information $Av_1=v_1,Av_2=4v_2.$ This gives you 2 equations on $a,b$ and 2 equations on $c,d.$ Solve. – Anne Bauval Jan 03 '23 at 11:00

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Hint

Rel the eigenbasis the matrix is diagonal (with the eigenvalues on the diagonal). Call it $D$.

Form the transition matrix: $$P=\begin{pmatrix}2\quad 1\\3\quad 2\end {pmatrix},$$ whose columns are the eigenvectors.

Compute $$P^{-1}DP.$$

calc ll
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