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 $x_3$.

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 $\text{Row} \; A$.

(a) A basis for $\text{Row} \; A$.

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 $\vec{x} \in \mathbb{R}^2$.

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
$p(x) = x^2 \quad \text{and} \quad q(x) = 1 + x$

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

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.

(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.

(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 $\vec{v}_3 = \left[ \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right]$.

(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

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 $A$.

*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 $A$.

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

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

(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.

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

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

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

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.

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

(b) Find an eigenfunction for $X$.

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

Log in or sign up to see discussion or post a question.