The Finale

This is a nice list of rules that don’t make a whole lot of sense by themselves because they’re all very conceptual. To fix the vagueness let’s look at some examples of inner products.

The first and most obvious inner product that exists is the traditional dot product. This guy works on any finite-dimensional vectors and he’s cool cause you can use him without needing to know that there are other possible inner products. The formula for the dot product is as follows:

If we have 2 n-dimensional vectors [3], u and v, where
u = (u_1, u_2, u_3, …, u_n) and
v = (v_1, v_2, v_3, …, v_n) then the dot product of u and v is

= u_1 * v_1 + u_2 * v_2 + u_3 * v_3 + … + u_n * v_n

With the dot product (and any inner product generally) we can take any two vectors we want and condense them down into a single number. This number returned from the dot product of two vectors is helpful most of the time for determining the angle between two vectors in higher-dimensional plans and blah blah blah. What I want to talk about isn’t what the inner products can be used for, it’s why you would need any inner products other than the dot product.

In the first linear algebra class, I took we never learned about the general body of inner products, we were just taught the dot product, shown how it worked, and then moved on to the next lesson. At the time this made sense to me because you can only really do so many operations on vectors that make sense, you can add them, subtract them, and with this, you can multiply them, and it doesn’t seem like it makes sense to try and do multiplication any other way, especially when doing it a different way would still lead you to the same answer. However, that’s not thinking big enough, because there are in fact vectors you can’t take the dot product of. Namely, you can’t take the dot product of infinite-dimensional vectors.

Infinite dimensional vectors are wacky and they can obviously be a pain because you have to come up with a whole new method for taking their inner product that follows all the rules while also being applicable to infinity. Very strange and very hard to do on the fly.

[1] A scalar is just like a regular number that isn’t a vector, like 1, 0.5, or pi. F is kinda like a general term for a space containing any scalar you might need (including imaginary scalars).

[2] This fifth rule is different when we’re working with imaginary numbers, but I’m just gonna pretend those don’t exist cause it’s not worth adding a little caveat to everything I say for them.

[3] I’m realizing I might not have explained what a vector is. Basically, an n-dimensional vector is just a list of numbers that is n numbers long. So a 3-dimensional vector could be (1, 2, 3) or (8, 27, 19).