MATH 54

Final | Spring '14 | Vojta

1. (20 points) Let $(x_1, x_2, x_3)$ be the solution to the linear system $\begin{align} x_1 + 2x_2 + 7x_3 &= 6 \\ 2x_1 + x_2 &= 4 \\ -x_1 + 3x_2 + 5x_3 &= 0 \end{align}$ Use Cramer's rule to find $x_3$.

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2. (30 points) Let @A = [[2, 4, 3, 1, 17], [1, 2, 0, 1, 2], [2, 4, 1, 2, 8], [2, 4, 2, -5, 19]]@ and @B = [[1, 2, 0, 1, 2], [0, 0, 1, 0, 4], [0, 0, 0, -1, 1], [0, 0, 0, 0, 0]]@. Given that $A$ is row equivalent to $B$, find:
(a) A basis for $\text{Row} \; A$.

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(b) A basis for Col $A$.

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(c) A basis for Nul $A$.

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3. (20 points) Given bases $\mathcal{B} = \left\{ \left[ \begin{array}{c} 1 \\ 1 \end{array} \right], \left[ \begin{array}{c} 1 \\ 2 \end{array} \right] \right\} \quad \text{and} \quad \mathcal{C} = \left\{ \left[ \begin{array}{c} 1 \\ 3 \end{array} \right] , \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \right\}$ find a matrix $M$ such that $[\vec{X}]_\mathcal{B} = M[\vec{x}]_\mathcal{C}$ for all $\vec{x} \in \mathbb{R}^2$.

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4. (20 points) Let $V$ be $\mathbb{P}_2$, with the inner product $< f, g > = \int_0^1 f(x)g(x) dx$ Compute the orthogonal projection of $p$ onto the subspace spanned by $q$, where $p(x) = x^2 \quad \text{and} \quad q(x) = 1 + x$

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5. (20 points) Find all least-squares solutions to the linear system $\begin{align} x_1 + 2x_2 + x_3 &= 0 \\ x_1 - x_3 &= 1 \\ x_2 + x_3 &= 1 \end{align}$

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6. (20 points) For each of the parts listed below, either give an example of such a matrix, or give a brief reason why no example exists. If you give an example, it must be either a specific matrix, or a matrix expression (involving sums, products, inverses, etc.) that evaluates to a specific matrix.
(a) A $3 \times 3$ matrix $A$ with eigenvalues $1, 2$, and $3$ and corresponding eigenvectors $\vec{v}_1 = \left[ \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right], \vec{v}_2 = \left[ \begin{array}{c} 2 \\ 1 \\ 0 \end{array} \right]$, and $\vec{v}_3 = \left[ \begin{array}{c} 3 \\ 4 \\ 2 \end{array} \right]$, respectively.

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(b) Same as part (a)< but with $\vec{v}_1 = \left[ \begin{array}{c} 0 \\ 1 \\ 2 \end{array} \right], \vec{v}_2 = \left[ \begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right]$ and $\vec{v}_3 = \left[ \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right]$.

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(c) Same as part (a), but with $\vec{v}_1 = \left[ \begin{array}{c} 1 \\ 1 \\ 0 \end{array} \right], \vec{v}_2 = \left[ \begin{array}{c} 1 \\ -1 \\ 1 \end{array} \right]$, and $\vec{v}_3 = \left[ \begin{array}{c} 2 \\ 0 \\ 1 \end{array} \right]$, and requiring that $A$ be symmetric.
For symmetric matrices, eigenvectors of distinct eigenvalues must be orthogonal. However, $\vec{v}_1$ and $\vec{v}_3$ are not orthogonal, so there can be no such matrix $A$.

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7. (25 points) (a) Compute the Wronskian $W[x, e^x, \sin x]$.

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(b) Are the function s$x, e^x, \sin x$ linearly independent? Explain, using the Wronskian.

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(c) Use a property of Wronskians to show that there is no differential equation $y''' + p_1y'' + p_2y' + p_3y = 0$ with $p_1, p_2$, and $p_3$ continuous on $(-\infty, \infty)$ for which $x, e^x$ and $\sin x$ are all solutions.

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8. (20 points) Let $A = \left[ \begin{array}{c c} 2 & 3 \\ 0 & 2 \end{array} \right]$ (a) Compute $e^{At}$.

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(b) Write down the fundamental matrix $X(t)$ for the differential equation $\vec{x}' = A\vec{x}$.

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(c) Write a general solution to $\vec{x}' = A\vec{x}$ as a linear combination of vectors.

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9. (30 points) (a) For the initial-boundary value problem

carry out separation of variables to produce two ordinary differential equations and all corresponding boundary or initial conditions.

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(b) Find an eigenfunction for $X$.

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10. (20 points) Find a formal solution to the initial-boundary value problem


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