"Let me show you... the reason eigenvalues were created, invented, discovered was solving differential equations, which is our purpose." - Gilbert Strang
For a course called "ODEs", the first month+ has required significantly more knowledge from undergrad linear algebra than undergrad differential equations.
I have a stronger understanding of why some universities combine calculus, linear, and DE.
And I'm moving from, "that sounds really cool!" to "trying to treat these concepts as disparate is like trying to separate pair bonded kittens" and "I have suffered a mathematical trauma from which I may never truly recover".
It's been a while since I was an undergrad... and I am suffering. It would be fun suffering if I had time to savor the learning, but alas.
Here's an incomplete list of things I probably should have known before taking this class. I've included some resources that helped with the ones that I really, really (, REALLY) needed to review.
Prerequisite Knowledge/Skills
This is not comprehensive & a work in progress. It reveals more about my foundational gaps than the course material.
Algebra/Precalculus
Complex numbers
Complex conjugate
Calculus
Fundamental Theorem of Calculus
Common Taylor Series expansions [ e^x, sin(x), cos(x), 1/(1-x) ]
Slope Fields (Direction Fields)
Resource:
UCSC - Commonly Used Taylor Series
It's what showed up first when I googled, "common taylor sereies expansions" and it does the trick if you need to reference the formulas.
Linear Algebra
Matrix Multiplication
Diagonal Matrix
Trace
Determinants
Identity Matrix
Invertible Matrices (and their Inverse)
Basis
Eigenvalues
Eigenvectors
Eigenspace
Subspace
Nullspace
Block Matrix
Jordan normal form (Jordan canonical form)
Resources:
3Blue1Brown - Essence of Linear Algebra <- The whole series
Organic Chemistry Tutor - How to Multiply Matrices - Quick & Easy!
Organic Chemistry Tutor - How to Find the Determinant of a 4x4 Matrix
MIT OpenCourseWare (Gilbert Strang) - Eigenvalues and Eigenvectors
^Where I got the quote at the top of this post!
The Bright Side of Mathematics - Jordan Normal Form 1
The Bright Side of Mathematics - Jordan Normal Form 2
Slope Field (Direction Field) <- sometimes this lives in calc, sometimes DE
Separation of Variables
Integrating Factor Method
Suggested Resources:
Mostly, I pulled out my dusty undergrad DE textbook. I suspect whatever DE textbook you have access to would be similarly helpful. If not,
Paul's Online Math Notes - Differential Equations
3Blue1Brown - Differential equations, a tourist's guide
Real Analysis
Norms
Limits & Convergence
Open and Closed sets
Compact Sets
Weierstrass M-Test
Lipschitz Continuity (uniformly, locally)
Resource:
I took a real analysis course recently, so I haven't used many resources to review. However, I really enjoyed this video:
CModGamma - Pointwise and Uniform Convergence
It's the only video on their channel. I can't explain how much it helped me understand pointwise and uniform convergence. Thank you, mysterious person!
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