Homework 1

1. Solving Linear Systems
- 1.1. (5 points)
- 1.2. (5 points)
2. Row Reductions
3. Interpreting RREFs
4. Embezzlement Charges

The following assignment is due Thursday 9/12 by 11:59 PM. You should submit all your written solutions to Gradescope as a single pdf.

Your solutions must be exceptionally neat and the final answer in your solution to each problem must be abundantly clear, e.g., surrounded in a very visible box. The graders have license to dock points for illegible or unclear solutions.
For the written part, choose the correct pages corresponding to each problem in Gradescope. Note that Gradescope registers your submission as soon as you submit it, so you don't need to rush to choose the pages. You will recieve no credit if you do not choose the correct pages, no exceptions.

1. Solving Linear Systems

For each of the following systems of linear equations, write down its augmented matrix and find its solutions using row operations. You must show your work, including each intermediate matrix in the reduction and the row operations you used.

Also write the solution to each system. Hint: There is a unique solution in each case.

1.1. (5 points)

\begin{align*} 2x + 4y &= 29 \\ -6x - 10y &= -75 \end{align*}

1.2. (5 points)

\begin{align*} x + 3y - 3z &= -2 \\ 2x + 7y + 7z &= -5 \\ x - 2y - z &= 3 \end{align*}

2. Row Reductions

(10 points) Find a sequence of elementary row operations (i.e., interchanges, scalings, and replacements) from the left matrix to the right matrix below. Show each of your steps, including each intermediate matrix and each row operations used.

\begin{align*} \begin{bmatrix} 1 & 2 & -1 & 5 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 5 \\ 2 & 2 & 1 & 1 \end{bmatrix} \sim \begin{bmatrix} 3 & 4 & 0 & 6 \\ 4 & 4 & 2 & 2 \\ 0 & -1 & 0 & 4 \\ 0 & 1 & 0 & 1 \end{bmatrix} \end{align*}

3. Interpreting RREFs

For each of the following systems of linear equations, write down

its augmented matrix
its reduced echelon form
whether it has
- no solutions
- a unique solution
- infinitely many solutions

You do not need to show your work for how you got the RREF. Hint: You can use a.rref() from SymPy.

3.1. (5 points)

\begin{align*} x_1 - 2x_2 + 5x_3 &= 6 \\ -2x_1 + 6x_2 - 11x_3 + 7x_4 &= -8 \\ 5x_1 - 10 x_2 + 25x_3 + 3x_4 &= 30 \end{align*}

3.2. (5 points)

\begin{align*} x_1 - 3x_2 + 4x_3 &= 0 \\ -x_1 + 6x_2 + x_3 &= 4 \\ 23x_2 + 5x_3 &= 9 \end{align*}

3.3. (5 points)

\begin{align*} x_1 + 5x_2 + 3x_3 &= -4 \\ x_1 + 6x_2 + 7x_3 &= -13 \\ -2x_1 - 12 x_2 - 14 x_3 &= 25 \end{align*}

4. Embezzlement Charges

(15 points) You're investigating a claim that, out of five competing car companies who purchase engine parts from the same suppliers, one purportedly overestimated their total spending by $1,300,000. Companies are required to report their total spending, and you've been able to determine how many units (× 10,000) of each part that each company has purchased. You haven't been unable to determine how much each item costs per unit (it's an industry secret), but you can assume that each company pays the same amount per unit. Given the following data, which company is falsifying their records? Explain your answer. Hint: Think of table below as a system of linear equations. How would you interpret a solution? You may (and probably should) use a.rref() from Sympy. (Errata 9/6 12:15PM There was an error in the wording of this problem and the associated matrix.)

Company	Part A	Part B	Part C	Part D	Part E	Total (× $100,000)
Company A	15	3	24	46	182	1013
Company B	5	3	14	25	100	552
Company C	15	3	24	46	188	1038
Company D	5	0	5	10	40	225
Company E	15	3	26	47	190	1056