Elementary Linear Algebra -- Spring 2018 -- final exam

MATH 329 -- Spring 2018 -- final exam

Name:_________________________ $\qquad$ 9 problems, 10 points each

• Write the problems in the given order, each problem on a separate page.
• Show enough work to justify your answers.

$\bullet$ 1. (a) Set up, but do not solve, a system of linear equations for the following problem: “Suppose your portfolio is worth $\$ 1,000,000$ and it consists of three stocks $A$, $B$, and $C$. The stocks are priced as follows: $A$ $\$2.10$ per share, $B$ $\$1.50$ per share, and $C$ $\$.50$ per share. Suppose also that you have twice as much of stock $A$ than $B$. How much of each do you have?” (b) Provide the augmented matrix and the matrix equation for this system. (c) How many possible answers are there?

$\bullet$ 2. Find the row-echelon form of this matrix: $$\left[\begin{array}{cc|c} 1&-1&-1\\ 2&1&0\\ \end{array}\right].$$

$\bullet$ 3. Suppose we have these two vectors in ${\bf R}^n$: $E_1=<1,0,0,...,0>$ and $E_2=<0,1,0,...,0>$. (a) What is the angle between them? (b) What is the magnitude of the vector $E_1-E_2$?

$\bullet$ 4. What is the matrix of the linear operator that stretches ${\bf R}^3$ by a factor of $2$ and flips it about the $xy$-plane?

$\bullet$ 5. Find the vector equation of the plane perpendicular to the vector $V=<1,2,3>$ and passing through $(2,3,1)$.

$\bullet$ 6. Find the eigenvalues of the linear operator given by the following matrix: $$F=\left[\begin{array}{cc} -1&-2\\ 1&-4\\ \end{array}\right].$$

$\bullet$ 7. The eigenvalues of the matrix given below are $0$ and $5$. Find one of its eigenvectors. $$F=\left[\begin{array}{cc} 1&2\\ 2&4\\ \end{array}\right].$$

$\bullet$ 8. Give the definitions of the image and the kernel of a linear operator. Find the image and the kernel for the following transformations of the plane: (a) rotation around the origin, (b) flip about the diagonal, (c) the composition of the stretch by a factor of $2$ and the projection on the $x$-axis.

$\bullet$ 9. Suppose $2\times 2$ matrices $A$ and $B$ satisfy: $AB=I_2$, where $I_2$ is the identity matrix. What is the relation between $A$ and $B$? Explain.