MATH 54

Midterm 1 | Fall '16 | Lin

1. Let $\vec{a}_1 = \left[ \begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right], \vec{a}_2 = \left[ \begin{array}{c} 1 \\ 1 \\ -1 \end{array} \right], \vec{a}_3 = \left[ \begin{array}{c} 1 \\ -1 \\ 1 \end{array} \right]$ Do the columns of $A = [ \vec{a}_1 \quad \vec{a}_2 \quad \vec{a}_3]$ span $\mathbb{R}^3$? Justify your answer.

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Are the columns of $A$ linearly independent? Justify your answer.

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Represent $\vec{b}$ as the linear combination of $\vec{a}_1, \vec{a}_2, \vec{a}_3$. Is this representation unique? Justify your answer. $\vec{a}_1 = \left[ \begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right], \vec{a}_2 = \left[ \begin{array}{c} 1 \\ 1 \\ -1 \end{array} \right], \vec{a}_3 = \left[ \begin{array}{c} 1 \\ -1 \\ 1 \end{array} \right], \vec{b} = \left[ \begin{array}{c} 1 \\ 2 \\ 1 \end{array} \right]$

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2. True or False: If True, explain why. If False, give an explicit numerical example for which the statement does not hold.
(a) $A \in \mathbb{R}^{n \times n}$ is invertible, then $A^{-1}$ is invertible.

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(b) If $n$ vectors in $\mathbb{R}^m$ are linearly dependent, then any vector can be represented by the linear combination of other $n - 1$ vectors $(n > 1)$.

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True/False: $A \in \mathbb{R}^{n \times n}$, then $(A^T)^2 = (A^2)^T$

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(d) Every subspace of $\mathbb{R}^n$ contains at most $n$ vectors.

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(e) Let $\vec{v}_1, \vec{v}_2, \vec{v}_3 \in \mathbb{R}^n$. If $\vec{v}_1$ and $\vec{v}_2$ are linearly dependent, then $\vec{v}_1, \vec{v}_2, \vec{v}_3$ are linearly dependent.

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3a) Compute $C = A^TB$, where $A = \left[ \begin{array}{c c c} 2 & 3 & 4 \\ 0 & 1 & 1 \end{array} \right], \quad B = \left[ \begin{array}{c c} 1 & 2 \\ 2 & 1 \end{array} \right]$

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b) Compute the matrix inverse of $A = \left[ \begin{array}{c c c} 1 & 0 & 0 \\ 0 & 2 & 1 \\ 0 & 1 & 1 \end{array} \right]$

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4. A linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^3$ has the following effect $T \left( \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] \right) = \left[ \begin{array}{c} 1 \\ -1 \\ 0 \end{array} \right], T \left( \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] \right) = \left[ \begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right]$ (a) Compute $T \left( \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] \right), T \left( \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \right)$

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(b) Write down the standard matrix of $T$, denoted by $A$.

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(c) Find a basis for the null space and column space of $A$.

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(d) Is $T$ injective? Is $T$ surjective? Justify your answer.

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(e) State the rank theorem, and verify the rank theorem for $A$ from the computation in (c).

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