Abstract: We'll look at some problems (in combinatorics and geometry) that are hard to solve head on. However, if we view the problems through the prism of linear algebra, we'll see that the solutions fall into place. The problems are just fun toy examples, but there is a serious point behind them---that linear algebra provides powerful tools that are widely applicable. Linear algebra, if not the key to life, is the key to much in mathematics, pure and applied. Please don't be scared off if you've not taken MATH 3333 (Linear Algebra): the ideas we'll use are intuitive, so the main thread of our discussion should still make sense.