Uploaded April 2021 | Updated September 2026, 1 week ago
This video has chapters to make the proof easier to follow.
Splitting explanation: youtu.be/ZINtBNje_08
In this video we give a proof of the classification theorem using two smaller proofs by induction. We show both the elementary divisor form and the invariant factor form of a module. This theorem tells us a lot about modules over a principal ideal domain!
Ring & Module Theory playlist: youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq7
0:00 First lemma
2:15 First lemma proof
8:51 Second lemma
9:24 Second lemma proof
11:20 Proof for p^r
28:30 Final proof
36:35 End of proof (additional explanation)
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Music: C418 - Pr Department
This video has chapters to make the proof easier to follow.
Splitting explanation: youtu.be/ZINtBNje_08
In this video we give a proof of the classification theorem using two smaller proofs by induction. We show both the elementary divisor form and the invariant factor form of a module. This theorem tells us a lot about modules over a principal ideal domain!
Ring & Module Theory playlist: youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq7
0:00 First lemma
2:15 First lemma proof
8:51 Second lemma
9:24 Second lemma proof
11:20 Proof for p^r
28:30 Final proof
36:35 End of proof (additional explanation)
Subscribe to see more new math videos!
Music: C418 - Pr Department
![How the Matrix Characteristic Polynomial is Connected to F[x]-Modules](https://i.ytimg.com/vi/jCt6mR3QtPk/mqdefault.jpg "How the Matrix Characteristic Polynomial is Connected to F[x]-Modules
Rational canonical form: https://youtu.be/q5uj4o0O5R0
Similar matrices isomorphism: https://youtu.be/-ligAAxFM8Y
Intro to F[x]-modules: https://youtu.be/H44q_Urmts0
The characteristic polynomial described by the determinant det(xI-A) is very useful when studying matrices. It turns out that its related to modules as well! Here we show how the characteristic polynomial is related to the module form of an arbitrary matrix.
Ring & Module Theory playlist: https://www.youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq7
0:00 Computing the determinant
10:07 Similar matrices have same polynomial
12:40 Proof for a general matrix
Subscribe to see more new math videos!
Music: C418 - Pr Department")
