Let Me Talk At You About Linear Algebra

Note: This is written off the top of my head, so keep that in mind.

Anyways. I was talking to my roommate (who is not in my field, but is learning introductory vectors for a class they are taking) and realized that I have. So many thoughts. About linear algebra. So allow me to talk at you about linear algebra.

Linear algebra education usually starts with learning about vectors. These are usually presented in a very physically-based situation (ex: physics, the vectors are first presented as "magnitude and direction," such as velocity or acceleration or displacement). However, this is simply a ruse to trick you into leaving the real world and entering The Vector Space.

Vector spaces are a type of mathematical conceptual space where all the objects are vectors. There are other such spaces, such as the space of all real numbers or the space of all integers or the space of all complex number or the space of all polynomials etc etc to infinity. In these spaces is where Math Happens. These spaces is why you can't, for example, add the integer 5 to the vector [1 1 0] -- they are in different spaces.

In a vector space, any vector has an address, like [1 1 0]. But this address is related to the coordinates you are using to view the vector in. Think of plotting a vector in x, y, z directions and then rotating everything and moving it to the left and drawing new a, b, c axes. This is called a Change of Basis. For a vector, a linear transformation of a vector is a Change of Basis. These linear transformations can also be written as matrices. Thus, you can understand a matrix as something with Matrix Properties as well as something with Linear Transformation properties. What's a linear transformation? Well, it transforms one element in a space into another element in another space. And it has certain properties which make it "linear."

Sidenote: I love how math just creates perfect little categories with perfect little rules and then is just logic puzzles between all the little categories. Well. If you look deep enough into math, then it falls apart because nothing is perfect in this world, but we can walk on the surface and pretend like nothing's wrong and that makes my little autistic brain so happy. My 10 year old self is so fucking happy right now, dude.

And then there's subspaces with their own properties (ex: image space, null space) and operators and dual spaces and guess what! You can turn a lot of other parts of math into linear algebra if you really want to (like signal processing?). And determinants and eigenvalues and eigenvectors.... And all of this eventually allows you to control a fucking robot! or calculate how a material will break! Wild.

Math, my favorite wizardry.