MATH 54

Midterm 1 | Spring '14 | Tataru

1. Consider the matrix $A = \left[ \begin{array}{c c c} 1 & 1 & 1 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \\ 0 & 2 & 4 \end{array} \right]$ Find rank $A$, a basis for $Col A$ and a basis for $Row A$.

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2. Compute
(a) $A^{-1}$, where $A = \left[ \begin{array}{c c c} 1 & 2 & 3 \\ 2 & 1 & 3 \\ 1 & 0 & 1 \end{array} \right]$

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(b) Compute $ABA$, where $A = \left[ \begin{array}{c c c} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} \right]$ and $B = \left[ \begin{array}{c c c} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} \right]$

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(c) Compute $\left[ \begin{array}{c c} 1 & 2 \\ 2 & 4 \\ 4 & 8 \end{array} \right] \left[ \begin{array}{c c c} 1 & 2 & 4 \end{array} \right]$

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(d) Compute $\det \left[ \begin{array}{c c c c c} 3 & 0 & 0 & 5 & 0 \\ 9 & 1 & 7 & 5 & 0 \\ 1 & 4 & 7 & 5 & 2 \\ 1 & 0 & 0 & 3 & 0 \\ 2 & 1 & 0 & 6 & 0 \end{array} \right]$

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3. (a) State Cramer's Rule.

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(b) Use it to solve the linear system (no credit for solving the system directly) $\begin{cases} x_1 + 2x_2 &= 7 \\ x_2 + 3x_3 &= 5 \\ x_1 - 2x_3 &= 3 \end{cases}$

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4. Mark each statement True or False. Justify your answers.
(a) If $AB = 0$ for two square matrices $A, B$, then either $A = 0$ or $B = 0$.

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(b) True/False: The set $P_2[X, Y]$ of all polynomials in $X$ and $Y$ of degree at most $2$ (together with the usual addition and multiplication by a constant) is a vector space of dimension $6$.

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5. Let $P_4$ denote the vector space of polynomial of degree at most $4$ (vector space together with the addition and multiplication by a constant). Consider the differentiation map $D: P_4 \to P_4$ given by $Df = f'$.
(a) Show that $D$ is linear.

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(b) Find a basis in $P_4$.

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(c) Find the matrix of $D$ in your chosen basis.

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True/False: If there is a linear transformation $T: \mathbb{R}^5 \to V$ which is onto, then dim $V \ge 5$.

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(b) Any linearly independent set in $\mathbb{R}^3$ must have exactly three elements.

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