Math 511: Linear Algebra

Final Exam: Review Guide

1. Consider the vector space \(\mathbb{P}_3\).

	a.)	Find the transition matrices between the ordered bases \(E = \left\{1,x,x^2\right\}\) and \(V = \left\{1,(x-3),(x-3)^2\right\}\).
	b.)	Find the matrix representation of the linear transformation \( D: \mathbb{P}_3 \to \mathbb{P}_3\) defined by \(D(p)(x) = p'(x)\) with respect to the basis \(V\).
	c.)	Find the matrix representation of the inner product on \(\mathbb{P}_3\) defined by \[\langle p,q \rangle = p(-1)q(-1) + \tfrac{1}{2}p(0)q(0) + p(1)q(1)\] with respect to the basis \(E\).
	d.)	Starting with the basis \(E\), apply the Gram-Schmidt algorithm to find an orthonormal basis for the inner product in part c.

2. Find the best least squares fit line to the data: \(\left\{ (-2,0), (-1,1), (0,4), (1,5)\right\}\).

3. Find all eigenvalues and corresponding eigenspaces of the matrix. \[ A = \left(\begin{array}{rrr} 4 & -5 & 1 \\ 1 & 0 & -1 \\ 0 & 1 & -1 \end{array}\right) \]

4. Consider the matrix \[ R = \left(\begin{array}{rr} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array}\right). \] Show that \(R\) will have complex eigenvalues if \(\theta\) is not a mulitple of \(\pi\). Give a geometric interpretation of this result.

5. You must be able to prove the following theorems for \(\mathbb{R}^n\) with the dot product.

	a.)	\(\mathbf{x}^T\mathbf{y} = \Vert\mathbf{x}\Vert\,\Vert\mathbf{y}\Vert \cos\theta \)    for \(\mathbf{x},\mathbf{y} \neq \mathbf{0}\).
	b.)	Cauchy-Schwarz-Bunyachevsky Inequality
	c.)	Triangle Inequality
	d.)	Parallelogram Law

6. Let \(\ell\) be the line \(y = mx\) in \(\mathbb{R}^2\), and let \(R_\ell :\mathbb{R}^2 \to \mathbb{R}^2\) be the linear transformation that reflects each vector in \(\mathbb{R}^2\) over the line \(\ell\). Find the matrix representation of \(R_\ell\).

7. Consider the linear transformation \(L :\mathbb{R}^3 \to \mathbb{R}^2\) defined by \[ L\left(\begin{array}{r} x_1 \\ x_2 \\ x_3 \end{array}\right) = \left(\begin{array}{c} -x_3 \\ x_1 - x_2 \end{array}\right). \] Find the kernel and image of \(L\).

8. Solve the linear system of differential equations. \[ \begin{cases} y_1' & = 2y_1 - 3y_2 \\ y_2' & = 4y_1 + 6y_2 \end{cases}\]

9. Solve the second order differential equation using matrix methods. \[ y'' - 5y' + 4y = 0 \]

10. Compute the exponential of the matrix, \(e^A\), for the matrix \[ A = \left(\begin{array}{rr} 1 & 2 \\ 0 & -1 \end{array}\right).\]

11. Apply the definition of the matrix exponential to compute \(e^A\) for the matrix \[ A = \left(\begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array}\right).\]

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