Due: October 6th, 2017

Math 18 Assignment 1

Answers to these problems are now available here. Note that these contain only the broad strokes of a complete solution, and not necessarily as much detail as you are expected to show.

  1. Go to Wolfram Alpha's System of Linear Equations Problem Generator (or another similar site of your choosing), and practice solving systems of linear equations until you can consistently solve them quickly and correctly. You don't need to submit anything, but the odds of you needing to solve at least one system of equations on the midterm or final are high.

    Do a similar thing for sums, differences, and scalings of vectors.

    You may also find it helpful to work several of the problems in the textbook.

  2. Determine which of the following are systems of linear equations in variables \(x_1,\) \(x_2,\) \(x_3,\) \(x_4,\) and \(x_5.\)
    1. \[\left\{\begin{array}{rcl}x_1 + x_5 &=& \sin(5) \\ x_4 + x_5 &=& \cos(5)\end{array}\right.\]
    2. \[\left\{\begin{array}{rcl}|x_1| + |x_2| &=& 2 \\ |x_3| + |x_4| + |x_5| &=& 3 \\ |x_1| + |x_5| &=& 1\end{array}\right.\]
    3. \[\left\{\begin{array}{rcl}0 &=& 0\end{array}\right.\]
    4. \[\left\{\begin{array}{rcl}x_1 + \cos\left(\frac{\pi}{13}\right)x_2 + \sin\left(\frac{\pi}{5}\right)x_3 &=& 1 \\ 444x_1 + 555x_3 - 523514x_5 &=& -415 \\ e^{\frac{\pi}{6}i}x_3 - e^{\frac{\pi}{173}i+5}x_4 &=& 4+12i\\ x_2 - x_5 &=& 4 + 3 + 2 + 1\end{array}\right.\]
    5. \[\left\{\begin{array}{rcl}x_3 &=& 3 \\ x_5 &=& 5 \\ x_2 &=& -18 \\ x_4 &=& 9 \\ x_5 &=& 92 \end{array}\right.\]
    6. \[\left\{\begin{array}{rcl}x_1 + x_3 + x_5 &=& 8 \\ x_2 + x_4 + x_6 &=& 9 \\ x_3 + x_5 + x_7 &=& 4\end{array}\right.\]
    7. \[\left\{\begin{array}{rcl}x_1 + x_2 &=& 5 \\ x_1 - 3x_2 &=& -1 \\ x_1 + x_2 &=& 5 \\ x_1 - 3x_2 &=& -1 \\ x_1 + x_2 &=& 5 \\ x_1 - 3x_2 &=& -1 \end{array}\right.\]
  3. Suppose that $(s_1, s_2, s_3)$ and $(t_1, t_2, t_3)$ are solutions to the linear equation $a_1x_1 + a_2x_2 + a_3x_3 = b$. Show that $\left(\frac{s_1+t_1}{2}, \frac{s_2+t_2}{2}, \frac{s_3+t_3}{2}\right)$ is a solution to the same linear equation.
  4. Write down three different augmented matrices for linear systems whose solution set contains only the point $(1,2,3)$.
  5. Which of the following matrices are in row echelon form? Of those, which are in reduced row echelon form?
    1. \[\begin{bmatrix} 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\\ \end{bmatrix}\]
    2. \[\begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\\ \end{bmatrix}\]
    3. \[\begin{bmatrix} 0 & 1 & 2 & 3\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 3\\ \end{bmatrix}\]
    4. \[\begin{bmatrix} 1 & 5 & 0 & 0 & 2 & 0 & 3\\ 0 & 0 & 0 & 1 & -4 & 0 & 8\\ 0 & 0 & 0 & 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{bmatrix}\]
    5. \[\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{bmatrix}\]
    6. \[\begin{bmatrix} 1\\ \end{bmatrix}\]
  6. Determine the solution sets for each of the following systems of linear equations in the variables $x_1,$ $x_2,$ $x_3,$ and $x_4.$
    1. \[\left\{\begin{array}{rcl}x_1 + x_2 + x_3 + x_4 = 4 \\ x_3 - x_4 = 0\end{array}\right.\]
    2. \[\left\{\begin{array}{rcl}x_1 + 2x_2 = 9 \\ -x_1 + 3x_2 = 6\end{array}\right.\]
    3. \[\left\{\begin{array}{rcl}x_1 + 2x_2 + 3x_3 = 4\\ 5x_1 + 6x_2 + 7x_3 = 8 \\ 9x_1 + 10x_2 + 11x_3 = 20\end{array}\right.\]
  7. Let $\vec a$ and $\vec b$ be the vectors given as follows: \[\vec a = \begin{bmatrix}2\\-1\\2\end{bmatrix},\qquad\qquad\vec b = \begin{bmatrix}-2\\1\\1\end{bmatrix}.\] For each of the following vectors, determine if it is in $W = \mathrm{span}\left\{\vec a, \vec b\right\},$ and express those that are as linear combinations of $\vec a$ and $\vec b.$
    1. \[\begin{bmatrix} 0 \\ 0 \\ 9 \end{bmatrix}\]
    2. \[\begin{bmatrix} 4 \\ -2 \\ 3 \end{bmatrix}\]
    3. \[\begin{bmatrix} 0.6 \\ -0.3 \\ 2.1 \end{bmatrix}\]
    4. \[\begin{bmatrix} -2 \\ 1 \\ 1 \end{bmatrix}\]
    5. \[\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}\]