MATH 54

Midterm 2 | Spring '16 | Wehrheim

1a) Find the change-of-coordinates matrix from the basis $\mathcal{B} = \{ 1, (1 + t)^2, (1 - t)^2, t^3 \}$ of $\mathbb{P}_3$ to the standard basis $\mathcal{C} = \{ 1, t, t^2, t^3 \}$.

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1b) Let $V$ be a finite dimensional vector space. State the definition for the dimension of a subspace $H$. Then explain which subspaces of $V$ have $\dim H = \dim V$.

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1c) State the definition of $\mathcal{B} = \{ b_1, \dots b_n \}$ being a basis of a vector space $V$.
Then use the properties in this definition to explain why the linear transformation $T: \mathbb{R}^n \to V$ given by $T(x) = x_1b_1 + \dots + x_nb_n$ is an isomorphism.

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2a) Let $A$ and $B$ be $3 \times 3$ matrices with $\det (A) = -1$ and $\det (B) = 3$. State appropriate properties of determinants to compute the following:
$\det(2A) = \dots$
$\det(BAB^T) = \dots$
The volume of the parallelepiped spanned by the columns of the matrix $B^{-1}A$ is $\dots$

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2b) Find (possibly complex) eigenvectors for each eigenvalue of $A = \left[ \begin{array}{c c c} 5 & 0 & -5 \\ 0 & 5 & 0 \\ 5 & 0 & 5 \end{array} \right]$.
(Hint: When calculating the characteristic polynomial, note a common factor that you should not multiply out. Then finding the roots only requires solving a quadratic equation.)

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2c) Suppose that $A$ is a $4 \times 4$ matrix with characteristic polynomial $\lambda(\lambda + \sqrt{5})(\lambda - \sqrt{7})^2$, and that $\mathbb{R}^4$ has a basis consists of eigenvectors of $A$. Specify a diagonal matrix $D$ that $A$ is similar to, state this similarity as a formula involving $A, D$, and another matrix $P$, and explain how to find $P$.

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3a) Find the general solution of $y^{(y)} + 4y^{(5)} - 3y^{(3)} - 18y' = 0$.
Then give an example of initial conditions that would specify a unique, nonzero solution.
You may use the identity $r^7 + 4r^5 - 3r^3 - 18r = r(r^2 - 2)(r^2 + 3)^2$.

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3b) Find the solution of $y'' + y' = t^2$ with $y(0) = 0$ and $y'(0) = 0$.

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3c) Find the general solution of $L[y] = \frac{1}{9}e^{-1} + 1$ and explain why there cannot be any other solutions, using only definitions and the following information (no theorems etc.):
1.) $\mathcal{L : C^{\infty} \to C^{\infty}}$ is a linear transformation,
2.) kernel $\mathcal{L} = \text{span} \; \{ e^{-t}, \cos 3t, \sin 3t \}$,
3.) $L[te^{-t}] = e^{-t} + 9$

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4a) Rewrite the ODE $y^{(4)} + 3y' + 5y = \cos 2t$ into a first order system for a vector function. Then assume that a solution of the system is given and explain how to obtain a solution $y$ of the ODE.

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4b) Find the solution of $x' = \left[ \begin{array}{c c} 0 & -1 \\ 1 & 0 \end{array} \right] x, \quad x(0) = \left[ \begin{array}{c} 0 \\ 1 \end{array} \right]$

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4c) The system $x' = \left[ \begin{array}{c c} 2 & 3 \\ 0 & 2 \end{array} \right] x$ has a solution $x_1(t) = e^{2t} \left[ \begin{array}{c} 1 \\ 0 \end{array} \right]$. Find a linearly independent solution by using the Ansatz ("educated guess") $x(t) = te^{2t} \left[ \begin{array}{c} 1 \\ 0 \end{array} \right] + e^{2t}v$ for an unknown vector $v$ in $\mathbb{R}^2$. Document your steps (plug in, solve, plug back).

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