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<p>The subject of linear algebra can be partially explained by the meaning of the two terms comprising the title. <q>Linear</q> is a term you will appreciate better at the end of this course, and indeed, attaining this appreciation could be taken as one of the primary goals of this course. However for now, you can understand it to mean anything that is <q>straight</q> or <q>flat.</q> For example in the $xy$plane you might be accustomed to describing straight lines (is there any other kind?) as the set of solutions to an equation of the form $y = mx + b$, where the slope $m$ and the $y$intercept $b$ are constants that together describe the line. In multivariate calculus, you may have discussed planes. Living in three dimensions, with coordinates described by triples $(x,\,y,\,z)$, they can be described as the set of solutions to equations of the form $ax+by+cz=d$, where $a,\,b,\,c,\,d$ are constants that together determine the plane. While we might describe planes as <q>flat,</q> lines in three dimensions might be described as <q>straight.</q> From a multivariate calculus course you will recall that lines are sets of points described by equations such as $x=3t4$, $y=7t+2$, $z=9t$, where $t$ is a parameter that can take on any value.</p>
+<p>The subject of linear algebra can be partially explained by the meaning of the two terms comprising the title. <q>Linear</q> is a term you will appreciate better at the end of this course, and indeed, attaining this appreciation could be taken as one of the primary goals of this course. However for now, you can understand it to mean anything that is <q>straight</q> or <q>flat.</q> For example in the $xy$plane you might be accustomed to describing straight lines (is there any other kind?) as the set of solutions to an equation of the form $y = mx + b$, where the slope $m$ and the $y$intercept $b$ are constants that together describe the line. If you have studies multivariate calculus, then you will have encountered planes. Living in three dimensions, with coordinates described by triples $(x,\,y,\,z)$, they can be described as the set of solutions to equations of the form $ax+by+cz=d$, where $a,\,b,\,c,\,d$ are constants that together determine the plane. While we might describe planes as <q>flat,</q> lines in three dimensions might be described as <q>straight.</q> From a multivariate calculus course you will recall that lines are sets of points described by equations such as $x=3t4$, $y=7t+2$, $z=9t$, where $t$ is a parameter that can take on any value.</p>
<p>Another view of this notion of <q>flatness</q> is to recognize that the sets of points just described are solutions to equations of a relatively simple form. These equations involve addition and multiplication only. We will have a need for subtraction, and occasionally we will divide, but mostly you can describe <q>linear</q> equations as involving only addition and multiplication. Here are some examples of typical equations we will see in the next few sections:
<p>This example is taken from a field of mathematics variously known by names such as operations research, systems science, or management science. More specifically, this is a perfect example of problems that are solved by the techniques of <q>linear programming.</q></p>
+<p>This example is taken from a field of mathematics variously known by names such as operations research, systems science, or management science. More specifically, this is a prototypical example of problems that are solved by the techniques of <q>linear programming.</q></p>
<p>There is a lot going on under the hood in this example. The heart of the matter is the solution to systems of linear equations, which is the topic of the next few sections, and a recurrent theme throughout this course. We will return to this example on several occasions to reveal some of the reasons for its behavior.</p>