Alternate coordinate systems (bases)

We explore creating and moving between various coordinate systems.

This is one of those tutorials that bring many ideas we've been building together into something applicable. Orthogonal projections (which can sometimes be conceptualized as a "vector's shadow" on a subspace if the light source is above it) can be used in fields varying from computer graphics and statistics! If you're familiar with orthogonal complements, then you're ready for this tutorial!

Finding a coordinate system boring. Even worse, does it make certain transformations difficult (especially transformations that you have to do over and over and over again)? Well, we have the tool for you: change your coordinate system to one that you like more. Sound strange? Watch this tutorial and it will be less so. Have fun!

As we'll see in this tutorial, it is hard not to love a basis where all the vectors are orthogonal to each other and each have length 1 (hey, this sounds pretty much like some coordinate systems you've known for a long time!). We explore these orthonormal bases in some depth and also give you a great tool for creating them: the Gram-Schmidt Process (which would also be a great name for a band).

Eigenvectors, eigenvalues, eigenspaces! We will not stop with the "eigens"! Seriously though, eigen-everythings have many applications including finding "good" bases for a transformation (yes, "good" is a technical term in this context).