Linear algebra
Alternate coordinate systems (bases)
We explore creating and moving between various coordinate systems.
Orthogonal complements
We will know explore the set of vectors that is orthogonal to every vector in a second set (this is the second set's orthogonal complement).
Orthogonal projections
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!
- Projections onto Subspaces
- Visualizing a projection onto a plane
- A Projection onto a Subspace is a Linear Transforma
- Subspace Projection Matrix Example
- Another Example of a Projection Matrix
- Projection is closest vector in subspace
- Least Squares Approximation
- Least Squares Examples
- Another Least Squares Example
Change of basis
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!
- Coordinates with Respect to a Basis
- Change of Basis Matrix
- Invertible Change of Basis Matrix
- Transformation Matrix with Respect to a Basis
- Alternate Basis Transformation Matrix Example
- Alternate Basis Transformation Matrix Example Part 2
- Changing coordinate systems to help find a transformation matrix
Orthonormal bases and the Gram-Schmidt Process
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).
- Introduction to Orthonormal Bases
- Coordinates with respect to orthonormal bases
- Projections onto subspaces with orthonormal bases
- Finding projection onto subspace with orthonormal basis example
- Example using orthogonal change-of-basis matrix to find transformation matrix
- Orthogonal matrices preserve angles and lengths
- The Gram-Schmidt Process
- Gram-Schmidt Process Example
- Gram-Schmidt example with 3 basis vectors
Eigen-everything
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).
- Introduction to Eigenvalues and Eigenvectors
- Proof of formula for determining Eigenvalues
- Example solving for the eigenvalues of a 2x2 matrix
- Finding Eigenvectors and Eigenspaces example
- Eigenvalues of a 3x3 matrix
- Eigenvectors and Eigenspaces for a 3x3 matrix
- Showing that an eigenbasis makes for good coordinate systems