Determinant Puzzle
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- Опубліковано 16 жов 2024
- Time for another matrix puzzle! Suppose f is a function from 2x2 matrices to a field that is multilinear, alternating, and satisfies f(I) = 1. What is f? In this video we'll show that f is none other than the determinant!
Generalization to n x n matrices: • Characterization of th...
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You gave the answer away in the title just a little bit :)
Just a little ;)
Using delta probably wasn't the best choice either xD
Zadig the guess is obvious, but is the proof?
K H the proof is trivial and left as an exercise for the viewer
I love the notation for "Want to find"
P.S.
You're the reason I enrolled in linear algebra this semester!
No way!!! Have fun 😄
@@drpeyam
thank you!
A good exercise for better understanding determinant. Very impressive, thank you!
Things get confusing really quickly when you switch to a different notation!
To clarify things a bit: start with
D(a b // c d)
= D( (0 + a.1) (b + a.0) // c d)
= D(0 b // c d) + a.D(1 0 // c d)
But
D(0 b // c d)
= D( (0 + b.0) (0 + b.1) // c d)
= D(0 0 // c d) + b.D(0 1 // c d)
But now notice that x = D(0 0 // c d) must be x = 0 as follows: since 0 = 0 + 1.0, you can write x as x = x + x; and since x is an element of a field (in particular, a group), we must have x = 0 by cancellation. (In fact, 0 is the only element of an (additive) group such that 0 + 0 = 0.)
The rest should be clear.
You really deserve to be a very good mathematician.
These videos help me to improve my mathematical analysis
Totally astonished at this approach. Cheers buddy
You are so inspiring to study math...awesome
@Dr.peyam how to integrate (x^2 cos 2x )/(4+sin^2 (2x)) dx...help me
Your videos are very useful for me
You are my inspiration
That's funny. On some of my classes, the generalization of that was considered definition of determinant and with my friends, we decide to call it 'the only reasonable definition' just for jokin'.
It's so powerful; great properties people forget about while studying - probably that's why our Profesor decided to make it a definition - so that everyone remembers :D
Cool! Awesome as usual!
Nice lecture sir
Where did this come from? It's beautiful😎
Sometimes this is how people define the Determinant, but I got that problem from the Friedberg textbook
@@drpeyam interesting, thanks for sharing !
Can we generalize to N dimensions by applying induction?
Characterization of the determinant ua-cam.com/video/AVMym1KfVXc/v-deo.html
Thanks
I didn't got the I (Identity Matrix) on the Thumbnail, because in Germany we use E (Einheitsmatrix)
I like that!
7:16 wtf??
Grazer Want To Find :)
@@drpeyam I was just kidding, love your videos btw Peyam :D
This problem is too easy to guess brainlessly. Aside from the determinant, what else do you know that takes matrices to reals?
Trace, or any entry of the matrix, or Trace A^2, etc.
@@drpeyam It appears that I was too brainless there lol.
Hahaha