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# Elementary Linear Algebra -- Spring 2018 -- midterm

Name:_________________________ $\qquad$ 8 problems, 10 points each
$\bullet$ 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: $$\left\{\begin{array}{lll} x&-y&+z&=-1,\\ 3x&&+z&=2,\\ 2x&+y&+z&=1. \end{array}\right.$$