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Sok$\newline$MATH$2250$S$24$PS$5$.p...$\newline$Raney$\newline$Linear Algebra$\newline$Spring $2024$$\newline$Problem Set $5$$\newline$$1$. Show that $\mathbf{x}=\left[\begin{array}{c}-2 \\ 1 \\ 1\end{array}\right]$ is an eigenvector of $A=\left[\begin{array}{ccc}0 & 1 & -1 \\ 1 & 1 & 1 \\ 1 & 2 & 0\end{array}\right]$ and find the corresponding eigenvalue.$\newline$$2$. Show that $\lambda=i=\sqrt{-1}$ is an eigenvalue of $A=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]$, and find a corresponding eigenvector.$\newline$$3$. Let $A=\left[\begin{array}{ll}2 & 4 \\ 6 & 0\end{array}\right]$. Find the eigenvalues of $A$ and bases for the corresponding eigenspaces.$\newline$$4$. Let $A$ be an idempotent matrix (that is, $A^{2}=A$ ). Show that $\lambda=0$ and $\lambda=1$ are the only possible eigenvalues of $A$.$\newline$$5$. Let $A=\left[\begin{array}{ccc}0 & 1 & -1 \\ 1 & 1 & 1 \\ 1 & 2 & 0\end{array}\right]$$1$ be the linear transformation defined by$\newline$$$T(p(x))=x p^{\prime}(x)$$$\newline$(a) Which, if any, of the polynomials$\newline$$$p_{1}(x)=1, p_{2}(x)=x, p_{3}(x)=x^{2}$$$\newline$are in $A=\left[\begin{array}{ccc}0 & 1 & -1 \\ 1 & 1 & 1 \\ 1 & 2 & 0\end{array}\right]$$2$ ?$\newline$(b) Which, if any, of the polynomials$\newline$$$p_{1}(x)=1, p_{2}(x)=x, p_{3}(x)=x^{2}$$$\newline$are in $A=\left[\begin{array}{ccc}0 & 1 & -1 \\ 1 & 1 & 1 \\ 1 & 2 & 0\end{array}\right]$$3$ ?$\newline$$6$. Find the dimension of the vector space $A=\left[\begin{array}{ccc}0 & 1 & -1 \\ 1 & 1 & 1 \\ 1 & 2 & 0\end{array}\right]$$4$ and give a basis for $A=\left[\begin{array}{ccc}0 & 1 & -1 \\ 1 & 1 & 1 \\ 1 & 2 & 0\end{array}\right]$$5$.$\newline$$7$. Define a linear transformation $A=\left[\begin{array}{ccc}0 & 1 & -1 \\ 1 & 1 & 1 \\ 1 & 2 & 0\end{array}\right]$$6$ by$\newline$$$T(A)=A B-B A$$$\newline$where $A=\left[\begin{array}{ccc}0 & 1 & -1 \\ 1 & 1 & 1 \\ 1 & 2 & 0\end{array}\right]$$7$. Find a basis for $A=\left[\begin{array}{ccc}0 & 1 & -1 \\ 1 & 1 & 1 \\ 1 & 2 & 0\end{array}\right]$$2$.$\newline$(D)$\newline$Dashboard$\newline$Calendar$\newline$$A=\left[\begin{array}{ccc}0 & 1 & -1 \\ 1 & 1 & 1 \\ 1 & 2 & 0\end{array}\right]$$9$$\newline$To-do$\newline$Notifications$\newline$$\lambda=i=\sqrt{-1}$$0$$\newline$Inbox