The Big Theorem, Part I

Learning Objectives:
1) State the two major problems: one geometric, one algebraic
2) State the Big Theorem, which compares span, linear combinations, and the REF form of a matrix
3) Prove the Big Theorem formally

This video is part of a Linear Algebra course taught at the University of Cincinnati.

[griffinbur]

Truly excellent and far, far clearer than the much more popular Khan and 3B1B series.

[bairamosmanov]

Your explanations are very edible for unprepared minds. Thank you from Moscow. Great job!

[bclan]

I am starting from this video in the playlist, and very excited. I love it when math shows how things are equivalent.

[АлексГудев]

A geometric explanation for what you talk about at 11:40 is that if all column vectors have a zero at the N-th coordinate, then they are 'locked' on a single plane and cannot 'rise' from it to the 3d space.
More generally, for an N-dimentional space, if all vectors have a 0 at a fixed coordinate, then they only live in a space one dimention smaller, R^(n-1).
And any vector not lying on their plane is unreachable, because it lives in a higher dimention - that is, no solution for `x` will take you from A's column vectors to `d`.

[Spinelesscloth]

You are my actual professor for this class and I can't thank you enough. My "professor" does't even lecture.

[theritesh]

Well simple, ...and very easy to understand... Thank you sir for making Algebra simple.

[sudarshann]

Have no words ❤️, actually Im trying to understand his every bit word . For this video I took much time and I saw it 6 times . But worth doing it was really amazing

[lexixu]

Finally! From here it begins to be practical for exams!

[dlambert]

12:08 I wonder if we could make this part a little more rigorous
"if Ax = b has a solution for every b∈R^m then Rx = d has a solution for every d∈R^m".
The constant vectors b and d will not necessarily be the same due to of row operations.

I think this follows from the fact that row operations on b only scale or add multiples of other b rows, so the generality of the 'every' isn't lost.

[melvicybanez]

The proof for #4 is Proof by Contrapositive. That is, one way to prove that #3 => #4 is to prove that not #4 => not #3.

[kathirs]

It will be really great if you could post some quizzes for each videos to test the understanding professor!!

[maazahmedpoke]

just wondering if the proof will still hold if there is a free variable?

[samueltomjoseph]

THIS VIDEO IS WHAT MAKES YOU DIFFERENT FROM THE TYPICAL MATH TEACHER

[shreyasgosavi]

12:00 What about infinite solution case, in that case there will be no leading 1 in one of the column?

[ssmm]

im so confused what do the m and n stand for? I thought they meant m (rows) and n (columns)