Appendix A: Linear Algebra

In real life, there is most often more than one variable. We wish to organize dealing with multiple variables in a consistent manner, and in particular organize dealing with linear equations and linear mappings, as those are both rather useful and rather easy to handle. Mathematicians joke that And well, they (the engineers) are not wrong. Quite often, solving an engineering problem is figuring out the right finite-dimensional linear problem to solve, which is then solved with some matrix manipulation. Most importantly, linear problems are the ones that we know how to solve, and we have many tools to solve them. For engineers, mathematicians, physicists, and anybody else in a technical field, it is absolutely vital to learn linear algebra.

As motivation, suppose we wish to solve

$$\begin{array}{c}x-y=2,\\ 2x+y=4,\end{array}$$

for $x$ and $y$. That is, we desire numbers $x$ and $y$ such that the two equations are satisfied. Let us perhaps start by adding the equations together to find

$$x+2x-y+y=2+4,\text{or}3x=6.$$