Algebra 56 - A Geometrical View of Gauss-Jordan Elimination

Although Gauss-Jordan Elimination is typically thought of as a purely algebraic process, when viewed geometrically, this process is beautiful and amazing, providing insights into the underlying mechanisms of the matrix transformations which lead to the solutions of a system of linear equations. Since a system of linear equations in three variables is graphically represented by a collection of planes, following how these planes change their orientation with each row operation can give us an intuitive understanding of how the transformation to reduced row echelon form works.

[AbarajithanGnaneswaran]

Beautiful! To add to the video, the key intuition behind Gauss Jordon is as follows:

We want to find the point (solution) where these three planes (rows) intersect. For that, we keep rotating (pivot) each plane along the line of intersection of two planes (rows). Since such a line always goes through the point, the point (our solution) remained unchanged as we rotate the planes in every iteration. We keep rotating them until each plane is perpendicular to an axis, at which point, the intercept at the axis reads the solution.

[Thefare]

I don't understand why linear algebra courses are so poorly designed. Most of the information that is crucial for understanding the topic is omitted from books. Do they think we are dumb and incapable of understanding such simple concepts?

[shampooner]

6 years later and this is still stellar content. Well done sir!

[dionsilverman]

Great visualization. I spent so much effort in my linear algebra class trying to visualize all this stuff, and what was happening in the transpose space.

I think what would make this even better though, would be if the planes were coloured differently, and the rows of the matrix coloured matchingly, to make it easier to keep track of which planes were being added as they're being talked about. It would also be nice to see the normal vectors represented by the row entries. I think that that would make more apparent why row addition causes plane rotation.

[MingoMash]

This is gold! Fantastic! :D "Jævlig bra, " as some of us say in Norwegian! This is a super clear and simple illustrative animation, and it worked incredibly well!

[northingtonsclassroom]

The geometric visual helped me make sense of the algebra. A few light bulb moments for concepts that I knew but didn't think about for this. Thank you!

[Groundsquirrel]

Amazing! A new way of understanding Gauss-Jordan Elimination.

[brunomartel]

So simple, relevant, intuitive and empowering. I'm starting to believe that there is a global conspiracy to keep things hard on purposem so we become more manipulable. PLEASE ADD auto-generated subs, when you accelerate videos, the normal subs lag! Thanks!!

[jkimmels]

Incredible job imparting fundamental insights & understanding! On behalf of people who don't understand it until we see it, thanks so much for helping us see it & understand!

[lynn]

Wow, thank you so much for this step by step with the visualization. Linear algebra is so interesting

[doodelay]

This helped more than u can imagine! more even then 3blue1brown, will watch the other vids as well

[lectrix]

This video is amazing! The geometric interpretation that I asked from my Linear Algebra teacher but she was unable to articulate. Please continue to make Linear Algebra, and Abstract Algebra concepts visualized in this manner. Helps build intuition and deeper understanding behind what is being done with these mathematical objects.

[konnen]

on a journey to improve my linear algebra and Calc 3 foundations for my machine learning major and this is an absolute beauty of a content. Keep them coming

[kigormley]

This is a fantastic series of lectures. Thanks a lot.

[TheNetkrot]

thanks, this was actually the thing I needed to see. This adds everything up for me .... thanks again.

[jinyuchen]

Great visualization and prefect supplementary content regarding RREF for 3B1B's Essence of Linear Algebra!

[andrewmathematician]

Man, this is brilliant, thanks from Czechia🇨🇿

[dauletakhmetov]

Great content, thanks from Kazakhstan

[vaclavbarta]

Dude, this is like magic, thanks a lot!

[eerrkk]

awesome insight, but the animations are frustratingly slow. thank goodness for 1.5x speed