MATH 54

Midterm 1 | Spring '15 | Vojta

(8 points) Suppose $A$ is a $5 \times 3$ matrix and $\vec{b}$ is a vector in $\mathbb{R}^5$ with the property that $A\vec{x} = \vec{b}$ has a unique solution. What can you say about the reduced echelon form of $A$? Justify your answer.

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Express the matrix $A = \left[ \begin{array}{c c} 2 & 1 \\ 8 & 5 \end{array} \right]$ as a product of elementary matrices.

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(10 points) Compute the determinant $ \left| \begin{array}{c c c c c c c} 2 & 0 & 10 & 11 & 8 & 9 & 0 \\ 0 & 3 & 11 & 13 & 10 & 5 & 0 \\ 0 & 0 & 1 & 2 & 1 & 3 & 0 \\ 0 & 0 & 1 & 3 & 2 & 4 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 3 & 2 & 0 \\ 0 & 0 & 1 & 1 & 3 & 2 & 0 \\ 0 & 0 & 9 & 4 & 8 & 7 & 2 \end{array} \right| $

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(10 points) Let $W = \{ \vec{p} \in \mathbb{P}_3 : \vec{p}(1) = \vec{p}'(2) + \vec{p}''(3) \}$ Is $W$ a subspace of $\mathbb{P}_3$? Explain.

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(10 points) Use coordinate vectors to test whether the following set of polynomials spans $\mathbb{P}_2$. Justify your conclusion. $1 - t + 2t^2, \; 2 + 5t^2, \; t + t^2, \; 3 - 3t + 8t^2$

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