Uploaded March 2020 | Updated September 2026, 2 weeks ago
Proof of Euler's identity: youtu.be/vHA50BASpTM
Explanation of the formula for the determinant of a 2x2 matrix in linear algebra: why is it ad - bc? We'll start with a proof from polar coordinates, then a really cool connection to complex numbers!
If you're interested in more content related to vector calculus, you can check out my series on vector calculus here: youtube.com/playlist?list=PLug5ZIRrShJHgsWPng59fFFoqn183aO-1
New math videos every Monday and Friday. Subscribe to make sure you see them!
Proof of Euler's identity: youtu.be/vHA50BASpTM
Explanation of the formula for the determinant of a 2x2 matrix in linear algebra: why is it ad - bc? We'll start with a proof from polar coordinates, then a really cool connection to complex numbers!
If you're interested in more content related to vector calculus, you can check out my series on vector calculus here: youtube.com/playlist?list=PLug5ZIRrShJHgsWPng59fFFoqn183aO-1
New math videos every Monday and Friday. Subscribe to make sure you see them!
![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")
