Due: October 13th, 2017

Math 18 Assignment 2

Answers to these problems are now available here.

  1. Go to Wolfram Alpha's Vector Times Matrix Problem Generator (or another similar site of your choosing), and practice multiplying matrices and vectors until you can consistently do it quickly and correctly. You don't need to submit anything, but the odds of you needing to perform at least one computation of this type on the midterm or final are high.

    You may also find it helpful to work several of the problems in the textbook; in particular, practice switching between systems of linear equations, vector equations, and matrix equations.

  2. Let $A$ be the following $3\times3$ matrix: $$A = \begin{bmatrix}-3 & -9 & 8 \\ 3 & 3 & -1 \\ 1 & -1 & 2\end{bmatrix}.$$ Determine the set of vectors $\vec{b} \in \mathbb{R}^3$ for which the matrix equation $A\vec{x} = \vec{b}$ in the variable $\vec{x}$ has a solution.
  3. Suppose $T$ is the following $22\times14$ matrix: $$T = \begin{bmatrix} -56 & 59 & -97 & -62 & 72 & -82 & 7 & 60 & -79 & 6 & 51 & -14 & 2 & -21 \\ -22 & -85 & -84 & 88 & 19 & 41 & 18 & -29 & 39 & -25 & -78 & -95 & -90 & -71 \\ 15 & 86 & -49 & 16 & 22 & -65 & 85 & -76 & 76 & 10 & 89 & 62 & -37 & -50 \\ 21 & -12 & -60 & 42 & 44 & -45 & 24 & -19 & 37 & 69 & 5 & 46 & 47 & 25 \\ -99 & -38 & -93 & 79 & -94 & -44 & -47 & 11 & -26 & -53 & -4 & -7 & -20 & 94 \\ -23 & -18 & -89 & -59 & -91 & 96 & -24 & 27 & 57 & 26 & -9 & -41 & -81 & -2 \\ -17 & -98 & 90 & 29 & -1 & 0 & 4 & -52 & -36 & 71 & 80 & -61 & 33 & -28 \\ -16 & 93 & -64 & 81 & 78 & -48 & 77 & 99 & -72 & -31 & 100 & 35 & 56 & -100 \\ 45 & -73 & 63 & -57 & 9 & 54 & 74 & -80 & -77 & -34 & 98 & -35 & -75 & 53 \\ 31 & -43 & 50 & 48 & -87 & -96 & 67 & 13 & -69 & -15 & 23 & 8 & -58 & -46 \\ 58 & 32 & -51 & 30 & -3 & 34 & 1 & 36 & -42 & -55 & 87 & 84 & 20 & -70 \\ -42 & 55 & 34 & 58 & -52 & -18 & -15 & 5 & 49 & -95 & 70 & 19 & 59 & 39 \\ 2 & -48 & 12 & 24 & -54 & -36 & 27 & -10 & 53 & 98 & -7 & -45 & 43 & 23 \\ 80 & -72 & -99 & -57 & 36 & -53 & -58 & 95 & -5 & 30 & -87 & 17 & 31 & -30 \\ 76 & 47 & 82 & -11 & -37 & 73 & -86 & 29 & -60 & 48 & -24 & 28 & -67 & -85 \\ 38 & 37 & 7 & -100 & 77 & -25 & -1 & 46 & -14 & -31 & -74 & -75 & -78 & 4 \\ -41 & 65 & 91 & 13 & 100 & -68 & 18 & 94 & 84 & -97 & 74 & 68 & -38 & 66 \\ -84 & -23 & 71 & 86 & -76 & -12 & 45 & 51 & 81 & -77 & -90 & -92 & -35 & -29 \\ 11 & 69 & -70 & -88 & 40 & -56 & 44 & 50 & 67 & 87 & 16 & 21 & 0 & -21 \\ -17 & -96 & -3 & 99 & 25 & 79 & -89 & -40 & -9 & -33 & 85 & -51 & 89 & -28 \\ 3 & -71 & -32 & 56 & -44 & -65 & 42 & -55 & 41 & -19 & 54 & -13 & 72 & -98 \\ 35 & -2 & -79 & -20 & 60 & 14 & 52 & 88 & -16 & -22 & 90 & 57 & -27 & -63 \\ \end{bmatrix}.$$ Does the matrix equation $T$$\vec{x} = \vec{b}$ in the variable $\vec{x}$ have a solution for every $\vec{b} \in \mathbb{R}^{22}$?

    (Hint: you do not need to perform row reduction to solve this problem.)

  4. Consider the following system of linear equations: $$\left\{\begin{array}{rcl} 3x_1 + 6x_2 + 6x_3 + 14x_4 &=& 0 \\ -x_1 + 6x_2 -6x_3 -2x_4 &=& 0 \\ 2x_1 + 6x_2 + 3x_3 +10x_4 &=& 0 \end{array}\right..$$ Find a spanning set for the solution set of the system.
    1. Give a solution to the following system of equations in parametric form: $$\left\{\begin{array}{rcl} 5x_1+3x_2+7x_3-x_4&=&8\\-2x_1-x_2-3x_3+x_4&=&{\color{red}{-2}}\\2x_1+2x_2+2x_3+2x_4&=&8 \end{array}\right..$$
    2. Show that if $\vec{v}$ and $\vec{w}$ are solutions to the above system of equations, then $\vec{v}-\vec{w}$ is a solution to the homogeneous system $$\left\{\begin{array}{rcl} 5x_1+3x_2+7x_3-x_4&=&0\\-2x_1-x_2-3x_3+x_4&=&0\\2x_1+2x_2+2x_3+2x_4&=&0 \end{array}\right..$$
  5. Determine which of the following sets of vectors are linearly independent, and support your claim:
    1. $\left\{ \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right\}$
    2. $\left\{ \begin{bmatrix} 8 \\ 2 \\ 2 \end{bmatrix}, \begin{bmatrix} -6 \\ 4 \\ 4 \end{bmatrix}, \begin{bmatrix} 5 \\ -5 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ -6 \end{bmatrix} \right\}$
    3. $\left\{ \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ -2 \\ 3 \end{bmatrix}, \begin{bmatrix} 6 \\ -1 \\ 1 \end{bmatrix} \right\}$
    4. The circle $\{(x, y) \in \mathbb{R}^2 | x^2 + y^2 = 1\}$
  6. Give an example of three vectors $\vec{x}$, $\vec{y}$, and $\vec{z}$ in $\mathbb{R}^4$ such that the sets $\{\vec{x}, \vec{y}\},$ $\{\vec{y}, \vec{z}\},$ and $\{\vec{z}, \vec{x}\}$ are linearly independent, but the set $\{\vec{x}, \vec{y}, \vec{z}\}$ is not.