-<p>We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word <q>solve</q> tends to get abused somewhat, as in <q>solve this problem.</q> When talking about equations we understand a more precise meaning: find <em>all</em> of the values of some variable quantities that make an equation, or several equations, true.</p>

+<p>We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word <q>solve</q> tends to get abused somewhat, as in <q>solve this problem.</q> When talking about equations we understand a more precise meaning: find <em>all</em> of the values of some variable quantities that make an equation, or several equations, simultaneously true.</p>

</introduction>

<subsection acro="SLE">

<title>Systems of Linear Equations</title>

+<p>Our first example is of a type we will not purse further. While it has two equations, the first is not linear. So this is a good example to come back to later, especially after you have seen <acroref type="theorem" acro="PSSLS" />.</p>

<p>We are about to give a rather involved proof, so a discussion about just what a theorem really is would be timely. Stop and read <acroref type="technique" acro="T" /> first.</p>

-<p>In the theorem we are about to prove, the conclusion is that two systems are equivalent. By <acroref type="definition" acro="ESYS" /> this translates to requiring that solution sets be equal for the two systems. So we are being asked to show <em>that two sets are equal</em>. How do we do this? Well, there is a very standard technique, and we will use it repeatedly through the course. If you have not done so already, head to <acroref type="section" acro="SET" /> and familiarize yourself with sets, their operations, and especially the notion of set equality, <acroref type="definition" acro="SE" /> and the nearby discussion about its use.</p>

+<p>In the theorem we are about to prove, the conclusion is that two systems are equivalent. By <acroref type="definition" acro="ESYS" /> this translates to requiring that solution sets be equal for the two systems. So we are being asked to show <em>that two sets are equal</em>. How do we do this? Well, there is a very standard technique, and we will use it repeatedly through the course. If you have not done so already, head to <acroref type="section" acro="SET" /> and familiarize yourself with sets, their operations, and especially the notion of set equality, <acroref type="definition" acro="SE" />, and the nearby discussion about its use.</p>

+

+<p>The following theorem has a rather long proof. This chapter contains a few very necessary theorems like this, with proofs that you can safely skip on a first reading. You might come back to them later, when you are more comfortable with reading and studying proofs.</p>

## src/section-SSLE.xml