Finally, recall the commutative diagram that helps organize...

Finally, recall the commutative diagram that helps organize our understanding of different bases (let $a$ denote the standard basis, while $p$ denotes the basis using the stretch directions (eigenvectors) of $A$). $A$ $\vec{x}_a$ $\rightarrow$ $A\vec{x}_a = \vec{y}_a$ $P$ $P$ $\vec{x}_p$ $\rightarrow$ $D\vec{x}_p = \vec{y}_p$ $D$ From this diagram we can see that $A = PDP^{-1}$. Find the matrices $P$, $D$, $P^{-1}$ and verify this equation.

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Problem 3. Consider a 2 imes 2 matrix A that has an eigenvector [[2],[1]] with associated eigenvalue -3 , and an eigenvector [[-1],[1]] with associated eigenvalue of 9 . Determine the image of the vector vec(x)=[[1],[2]] under A in the following two ways: (a) Express vec(x) as a linear combination of [[1],[2]],[[-1],[1]] and use your knowledge of eigenvalues. (b) Calculate the matrix A and directly find Avec(x). Finally, recall the commutative diagram that helps organize our understanding of different bases (let alpha denote the standard basis, while ho denotes the basis using the stretch directions (eigenvectors) of A ). we can see that A=PDP^(-1). Find the matrices P,D,P^(-1) and verify this equation. Finally, recall the commutative diagram that helps organize our understanding of dif- ferent bases(let a denote the standard basis,while o denotes the basis using the stretch directions (eigenvectors) of A). A To Axa=Ja Tp D=Yp D this equation.

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