Why do we multiply matrices the way we do??

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[nutriascoc]

Learn english and discover your channel was the best thing ive ever done. In my native lenguage there are not many people interested in maths, at least in youtube.

[Ensivion]

As someone who had to discover this on my own, I have to say that this video pretty much should be a standard lesson to people who are new to linear algebra. It's just so simple and it needs to be drilled into students' heads. It was fun to discover myself, but the use of this simple motivation can inspire students to come up with extensions of their own. This lesson is dense and right to the point and is very foundational.

[rockysmith]

THIS. Just, THIS. I swear, if only this is how they introduced matrix multiplication it'd make so much more sense than just forcing it. Taking students through and seeing how substitution from one point to another point in a linear transformation sequence has this nice pattern, this pattern we now call matrix multiplication is simply wonderful.

[thatdude_]

The multiplication rule for matrices A, B: V —> V (for V a finite dimensional vectorspace) actually pops out quite naturally when you look at how A, B and AB act on a Basis of V and expand in terms of that basis

[MasterHigure]

This is the best way of thinking of matrix multiplication: It is really compositions of linear maps, and matrix multiplication is what it has to be so that the correspondence matrix <-> linear transformation works.

[bigjazbo]

Great video demonstrating the concept behind the mechanics of matrix multiplication. When I was an engineering student 45 years ago, I realized that the right-hand rule for vector multiplication and the row x column multiplication rules for matrices were arbitrary, so when I encountered them in engineering problems I'd often use left-hand rule and "reflected" matrix multiplication to get the exact same final physical answer (which raised some eyebrows of the teachers grading my submittals). I think a video explaining and generalizing our arbitrary rules would be interesting, but I've never seen such a thing in a book or on video.

[omograbi]

7:53 I noticed Z1 should be Z2 here, I was confused a little bit at first.

[quantumgaming]

This is indeed a great video. I would also recommend 3blue1brown's channel on this topic in his "Essence of Linear Algebra" that tries to naturally evoke all these concepts in a way that anyone could understand.

[goodplacetostop]

7:51 You meant z_2 ?
16:14 Good Place To Stop

[pierreabbat]

15:45 The traces of the two products should be equal, but they aren't. You changed the 1 in the lower right to a 3 when you put them in the opposite order.

[indivisible_vaughn]

Shouldn't the first entry of the 2x2 matrix be 23 instead of 21?

[lucaolmastroni]

Hi, at t.c 14:26, the value of row_1 column_1 should be 23 instead of 21 if I'm not wrong. As it is said in last lines of "Some like it hot" "nobody's perfect." Teacher's mistakes shortens the distance from the pupils and that's encouraging for them- 😅

[manucitomx]

That was very constructive.
Thank you, professor.

[otakurocklee]

Nice video, but wish you had mentioned "associativity". ie: we want A(Bx) = (AB)x. we want associativity, so given the definition of matrix-vector multiplication, and given we want matrix multiplication to be associative, we must define matrix-matrix multiplication this way. So the motivation is to make matrix multiplication associative.

[edwardlulofs]

You did something similar, but more basic, recently. I had never questioned the matrix multiplication algorithm before, so I was fascinated by your explanation. When you started this episode, I had only a vague memory of how you did the previous version. This told me that I hadn't mastered the material and needed review.
But then, you expanded the explanation to the tensor product and tensor contraction. Brilliant. I had never understood these concepts so clearly before.
Thank you very much. 😃

[leolucas]

2:21 - It would be probably good to explain to students why x is set as a column in this step, instead of an arrow, as the position of x1 and x2 in the equations would suggest). This would result in another rule for matrix multiplication.

I know we want to keep an analogy between the vectors x and y, but this may not be obvious.

[supersolvable]

Small typo in 7:55 you wrote z1 instead of z2

[iabervon]

If you swivel the vector counter-clockwise into the matrix rows instead of swiveling the rows into the vector, you do your calculations in the places where they would be with the separate equations, rather than sideways and on top of each other. Doesn't help for matrix multiplication, but it's easy enough to remember that as two columns next to each other that you do one at a time.

[PaulMurrayCanberra]

You see, that third row on the blackboard, I would have written it the other way around - (x1, x2) goes *through* the matrix, so the matrix belongs on the right: subject verb object. Matrix multiplication only made sense to me when I realised that the matrix was being treated like a *function*, and we write functions f(x) with the operator on the left and the operand on the right. Worked my way halfway through "the geometry of complex numbers" before I worked out what was going on and why my code wasn't doing the right thing.

[aaronclair]

This is awesome. I've previously tried and failed to build a physical intuition for matrix multiplication, and I think this is the first time it's really clicked for me. If you do it any other way, it basically doesn't work or it turns into a horrible mess, whereas this way is the only elegant way to nest systems of linear equations or linear maps inside one another. Excellent presentation, thanks.