Uploaded October 2024 | Updated September 2026, 1 week ago
We can write 1/(1-x) as the Taylor series 1+x+x^2+x^3+... But can we prove the simple identity d/dx (1/(1-x)) = 1/(1-x)^2 using only the infinite sum? The answer is yes, and it leads us to the more general idea of Cauchy products!
Calculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao
0:00 The Derivative
2:14 The Square
7:50 Generalization: Cauchy Products
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Music: C418 - Smooth Fall
We can write 1/(1-x) as the Taylor series 1+x+x^2+x^3+... But can we prove the simple identity d/dx (1/(1-x)) = 1/(1-x)^2 using only the infinite sum? The answer is yes, and it leads us to the more general idea of Cauchy products!
Calculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao
0:00 The Derivative
2:14 The Square
7:50 Generalization: Cauchy Products
Subscribe to see more new math videos!
Music: C418 - Smooth Fall
![F[x]-Module Derivation of Rational and Jordan Canonical Forms](https://i.ytimg.com/vi/q5uj4o0O5R0/mqdefault.jpg "F[x]-Module Derivation of Rational and Jordan Canonical Forms
Similar matrices isomorphism proof: https://youtu.be/-ligAAxFM8Y
Every module is a direct sum of cyclic modules: https://youtu.be/gWIRI43h0ic
Intro to F[x]-modules: https://youtu.be/H44q_Urmts0
The rational canonical form and Jordan normal form of a matrix are very important tools in linear algebra, but ring theory and module theory give us a very effective way to prove their existence! Here we show that every matrix is similar to a matrix in rational and Jordan canonical form.
Ring & Module Theory playlist: https://www.youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq7
0:00 Rational canonical form
14:17 Every matrix is similar to RCF
18:33 Algebraically closed fields
20:10 Jordan canonical form
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Music: OcularNebula - The Lopez")
