An Illustrated Introduction to Linear Algebra

Summary

This post assumes you know algebra, but no linear algebra. Lets dive in.There are two big ideas I want to introduce in the first chapter: Gaussian elimination, (which is not strictly a linear algebra thing, and has been around for years before linear algebra came along), and row picture versus column picture, which is a linear algebra thing.Let’s say you have a bunch of nickels and pennies, and you want to know how many of each do you need to have 23 cents.You could write that as an equation that looks like this:x is the number of nickels you need, y is the number of pennies you need. And you need to figure out the x and y values that would make the left-hand side work out to 23. And this one is pretty easy, you can just work it out yourself. You’d need four nickels and three pennies.So x is four, y is three.This kind of equation is called a linear equation. And that’s because when you plot this equation, everything is flat and smooth. There are no curves or holes. There isn’t a ^2 in the equation for example to make it curved. Linear equations are great because they’re much easier to work with than curved equations.Aside: Another solution for the above is 23 pennies. Or -4 nickels + 43 pennies. The point is you have two variables (x and y for nickels and pennies), and you are trying to combine them in different ways to hit one number. The trouble starts when you have two variables, and you need to combine them in different ways to hit two different numbers. That’s when Gaussian elimination comes in. In what world would you have to hit two different numbers? Does that seem outlandish? It’s actually very common! Read on for an example. Now let’s look at a different example. In the last one we were trying to make 23 cents with nickels and pennies. Here we have two foods. One is milk, the other is bread. They both have some macros in terms of carbs and protein:and now we want to figure out how many of each we need to eat to hit this target of 5 carbs and 7 protein.This...