Why Gaussian Elimination works

In this video, I show why the method of Gaussian elimination works, in the sense that why you won't gain or lose solutions when you row-reduce a matrix.

[zaraal-ghnai]

This guy asks brilliant questions and answers them brilliantly.

[frozenmoon]

A wise man said "It works!" and so began a new era of great math. :)

[davidgould]

I love the hyper-theoretical (possibly "grown up" is the proper adjective) explanation of what I always (probably mistakenly) always saw as obvious.

My take on Gaussian elimination was that you are doing what you're told to do when you're about 7 or 8 to solve simultaneous equations: add and subtract multiples of the equations until you've got rid of enough variables that you can solve for one, then substitute back in for the others. The only difference is that you save some ink by not writing x, y, z, etc every time and use this "matrix" notation.

Probably what this video is doing is proving that what I was doing 50 years ago actually *works* (eg no missed solutions). Or that I've missed something fundamental. I'll watch it again.

[dodarwin]

I've been wondering if gaussian elimination was just some meaningless crap that happened to work by chance. Now that I'm finally understand it thanks to this video.

[olz]

Hey. This video is great!
I was left wondering why the elementary matrixes and their product is invertible.
However, I think this might be because intuitively you can always undo the elementary row operations.

[toradey]

Sir your videos are just amazing.I am Subhrajit from India A student (in youtube) of yours. thank you sir.😀😀😀

[laiznogueira]

Nice video!!! The hardest to understand on my head still is, how adding one equation to other Works to gaussian elimination, it looks so crazy when I think about the equation as a real system representation. I'll look for a video about this

[HDQuote]

Dr. Peyam, I love your videos and this is not really related, but can you maybe do a video about your life so far?
I noticed you are from germany, but lived in france in your youth and you speak so many languages. This really got me interested to hear about what lead to this extraordinary development.

[pierreabbat]

Can you talk about numerical stability in matrix arithmetic? I wrote a test of a matrix class as follows:
1. Put 1 on all diagonal elements of a square matrix
2. Fill one triangle of the matrix with random numbers, leaving the other triangle 0
3. Permute the rows and columns, doing an even number of swaps
4. Check that the determinant is 1.
Sometimes it computes the determinant as slightly different from 1.

[azhar]

How many of you are left handed? I am.

[zaraal-ghnai]

Please more videos on abstract algebra.

[holyshit]

Yes but there are some cases where it is numerically unstable
When I used to go to high school we did not use matrix in Gaussian elimination

[almightyhydra]

det A = 0 - 1(6-4) + 2(4-3) = 0 which explains the solution having (at least one) free variable.

[cletusspuckler]

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