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# Elementary Linear Algebra -- Spring 2018 -- midterm

**MATH 329 -- Spring 2018 -- midterm exam**

Name:_________________________ $\qquad$ 8 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.** 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 two stocks $A$ and $B$. The stocks are priced as follows: $A$ $\$2.1$ per share, $B$ $\$1.5$ per share. Suppose also that you have twice as much of stock $A$ than $B$. How much of each do you have?”

$\bullet$ **2.** In an effort to find the line in which the planes $ 2x -y- z=2 $ and $-4x+2y+2z=1$ intersect, a student multiplied the first one by $2$ and then added the result to the second. He got $0=5$. Explain the result.

$\bullet$ **3.** Vectors $A$ and $B$ are given below. Copy the picture and illustrate graphically: (a) $A+B$, (b) $A-B$, (c) $||A||$, (d) the projection of $A$ on $B$, (e) the projection of $B$ on $A$.

$\bullet$ **4.** Find the angle between the vectors $<1,1,1>$ and $<1,0,0>$. Don't simplify.

$\bullet$ **5.** Find the vector equation of the line parallel to both $xy$- and $xz$- coordinate planes and passing through $(2,3,1)$.

$\bullet$ **6.** Solve the system of linear equations:
$$\left\{\begin{array}{lll}
x&-y&=-1,\\
2x&+y&=0.\\
\end{array}\right.$$

$\bullet$ **7.** Is it possible that a system of linear equations has (a) no solutions, (b) one solution, (c) two solutions, (d) infinitely many solutions? Give an example or explain why it's not possible.

$\bullet$ **8.** Represent the system of linear equations as a matrix equation:
$$\left\{\begin{array}{lll}
x&-y&+z&=-1,\\
3x&&+z&=2,\\
2x&+y&+z&=1.
\end{array}\right.$$