MATH 54

Midterm 2 | Fall '16 | Lin

1. (10 points) $A = \left[ \begin{array}{c c c} 2 & -1 & 0 \\ -1 & 3 & -1 \\ 0 & -1 & 2 \end{array} \right]$.
Find all its eigenvalues and the corresponding eigenvectors.

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2. True or False (15 points) If True, explain why. If False, give a counterexample. The correct answer is worth 1 point for each problem. The rest of the points came from the justification.
(a) (3 points) If $A \in \mathbb{R}^{n \times n}$ with $\det(A) = 3$, then $\det(A^{-1}) = \frac{1}{3}$.

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(b) (3 points) If $A \in \mathbb{R}^{n \times n}$ is an invertible matrix, and $A$ is diagonalizable, then $A^{-1}$ is diagonalizable.

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(c) (3 points) The vector spaces $\mathbb{P}_3$ and $\mathbb{R}^3$ are isomorphic.

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(d) (3 points) $V$ is a vector space. If $\dim V = n$ and $S$ is a linearly independent set in $V$, then $S$ is a basis for $V$.

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(e) (3 points) If $A, B \in \mathbb{R}^{2 \times 2}$ and both $A$ and $B$ are diagonalizable, then $AB$ is diagonalizable.

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3. (10 points) Let $\mathcal{B} = \{ \vec{b}_1, \vec{b}_2, \vec{b}_3 \} \equiv \{ 1, t - 1, (t - 1)^2 \}$ be a subset of $\mathbb{P}_2$.
(a) (5 points) Show that $\mathcal{B}$ is a basis for $\mathbb{P}_2$.

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(b) (5 points) Find the $\mathcal{B}$-coordinate of $1 + 2t + 3t^2$.

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4. (10 points) Let $\mathcal{B} = \left\{ \left[ \begin{array}{c} 7 \\ - 2 \end{array} \right] , \left[ \begin{array}{c} 2 \\ -1 \end{array} \right] \right\}, \mathcal{C} = \left\{ \left[ \begin{array}{c} 4 \\ 1 \end{array} \right] , \left[ \begin{array}{c} 5 \\ 2 \end{array} \right] \right\}$.
(a) (7 points) Find the change-of-coordinates matrix $P_{C \leftarrow B}$.

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(b) (3 points) Find $[x]_\mathcal{C}$, where $[x]_\mathcal{B} = \left[ \begin{array}{c} 4 \\ 5 \end{array} \right]$.

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5. (10 points) Consider the subspace $W$ of $\mathbb{R}^4$ spanned by $\vec{u} = \left[ \begin{array}{c} 1 \\ 0 \\ -2 \\ 2 \end{array} \right], \; \vec{v} = \left[ \begin{array}{c} 1 \\ -1 \\ 0 \\ 4 \end{array} \right]$ (a) (5 points) Find a nonzero vector $\vec{w}$ in $W$ that is orthogonal to $\vec{u}$.

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(b) (5 points) Find the orthogonal projection of the vector $\vec{y} = \left[ \begin{array}{c} 3 \\ -1 \\ 2 \\ 1 \end{array} \right]$ to the subspace $W$.

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6. (10 points) Let $T: \mathbb{R}^{2 \times 2} \to \mathbb{R}^{2 \times 2}$ be the linear transformation given by $T(A) = A^T$, where $A^T$ is the transpose of $A$.
(a) (2 points) Is $T$ an isomorphism? If so, write down $T^{-1}$.

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(b) (2 points) Find $[T]_\mathcal{B}$, which is the matrix representation o $T$ under the basis $\mathcal{B} = \left\{ \left[ \begin{array}{c c} 1 & 0 \\ 0 & 0 \end{array} \right] , \left[ \begin{array}{c c} 0 & 1 \\ 0 & 0 \end{array} \right] , \left[ \begin{array}{c c} 0 & 0 \\ 1 & 0 \end{array} \right] , \left[ \begin{array}{c c} 0 & 0 \\ 0 & 1 \end{array} \right] \right\}$

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(c) (6 points) Find the eigenvalues and the eigenspaces of $[T]_\mathcal{B}$.

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