TimeTravelPenguin
commented
2 years ago I highly recommend Gilbert Strang's text "Linear Algebra", as he has some very well structured knowledge on linear algebra, and, iirc, everything fits so nicely together in terms of required topics from the mathematics toolbox.
Additionally, I also recommend using the applications of solving Ordinary Differential Equations.
I just finished a course on ODEs, and the linear algebra of solving systems of linear ODEs is fascinating and has some incredible visual connections. One such case is visualising non-linear ODEs near extrema by solving linearised solutions about that point. If you give it some thought, perhaps that will nicely do justice to presenting your ideas at a high-level!
I'm a student who just got my undergraduate degree in math, and I've got an idea for an exciting video (maybe even a series). In 3B1B's video on eigenvectors, he covers the basic idea of vectors that stay on their span during a transformation and how to diagonalize a matrix. However, he mainly covers real and diagonalizable matrices, whereas the full theory of eigenvectors applies to many more transformations. There is a variation called Jordan-diagonalization which can be done to any matrix (even those without enough eigenvectors).
I'm looking for a video producer with whom I can share this understanding. I'm hoping we could find an excellent way to present all of this info with a discovery-based approach.
To motivate this concept, we could consider a vector v(t) that depends on time t and satisfies v'(t)=Mv(t), where M is a matrix. The solution to this problem is very understandable if M is diagonalizable. (Form the matrix e^(Mt) (it has the same eigenvectors as M, but the eigenvalues are e^(λt) instead of λ), and multiply it by v(0) (v(t) at time t=0)) However, if M is not diagonalizable, then understanding generalized eigenvectors is required to proceed.
I was thinking of covering: Complex eigenvalues and eigenvectors (there are some nice animations and 'a-ha' moments here) An understanding of sub-transformations (which is needed to tie everything together, and to understand Nillpotent matrices) Nilpotent matrices (A^n=0 this part helps motivate generalized eigenvectors.) Generalized eigenvectors and Jordan-block matrices (the key to our problem) A quick proof that every matrix can be Jordan-diagonalized (yes, all of them) And, to conclude the video (series), how to solve v'(t)=Mv(t).
Contact details:
Email: erikstephens98@gmail.com. You can email me if you want to collab or leave a comment here.