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Computational Linear Algebra: Notes and Questions for MATH 3325

Nicholas Long

Section 1.1 Warmup Problems

You should try to do the following activity right away.

Activity 1.1.1.

(a)

Solve:
\begin{align*} 3x_1-2x_2\amp= 6\\ -x_1+x_2\amp= 1 \end{align*}

(b)

Draw a graph of the solution set of the equation: \(3x_1-2x_2=6\text{.}\)
Hint.
If a solution has \(x_1=a\text{,}\) what is \(x_2\) or viceversa?

(c)

Draw a graph of the solution set of the equation: \(-x_1+x_2=1\text{.}\)

(d)

Graph the solution sets from the two previous steps together. How does your answer to part 1.1.1.a compare to your graph?

(e)

Solve:
\begin{align*} 2x_1-2x_2\amp=6\\ -x_1+x_2\amp=1 \end{align*}

(f)

Solve:
\begin{align*} 2x_1-2x_2\amp=-2\\ -x_1+x_2\amp=1 \end{align*}

(g)

Wait, what just happened? Explain the results of the previous two parts. What do the graphs of the corresponding solution sets look like in relation to the graphs of the equations?

(h)

What are the possible intersections of two lines? Clearly state your conjecture which adresses ALL possibilities.
Throughout this course we will be doing many of the same things you did in the previous questions, but we will do them in a more general setting that will allow us to solve many new and old problems.

Subsection 1.1.1 Application Warmup Problem

Here is an application problem where you can see the relevence of your work you just did.

Activity 1.1.2. Introduction to Traffic Flow.

Consider the following diagram of a network with the flows indicated:
Figure 1.1.1. A network with directions of flow
If all the flows, \(x_1\text{,}\) \(x_2\text{,}\) \(x_3\text{,}\) \(x_4\text{,}\) and \(x_5\text{,}\) are all nonnegative, what is the largest possible value for \(x_3\text{?}\)
Hint.
To answer this consider the following:
  1. For each node (A, B, C, D) set up a conservation equation. Remember the flow in must equal the flow out.
  2. How many linear equations do you have? How many unknowns?
  3. Use your conservation equations for A and D to solve for \(x_3\) in terms of \(x_1\) and \(x_5\text{.}\) How might we solve for \(x_2\) and \(x_4\) in terms of \(x_3\) and \(x_5\text{?}\)
  4. Is our solution unique?
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