Matrix multiplication as composition | Chapter 4, Essence of linear algebra

Multiplying two matrices represents applying one transformation after another.
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Spanish: Juan Carlos Largo
Vietnamese: @ngvutuan2811

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[akmalrazak]

Learning matrix in high school was like learning how to construct a sentence but never know it was for communication

[wertnick]

20 years after first being exposed to matrices, and getting a computer engineering degree, I finally understand matrix multiplication.

[GreenDayxRock]

So for everyone else that had trouble with his "honest to god proof"... I've taken a number of courses in linear algebra and many proof courses, and found the same hole in his explanation. Here's what he means though, he just left out a crucial intermediary in his proof:

A(BC) means apply the overall effect of BC and then A. Of course, the overall effect of BC is equivalent to applying C then B based on what he explained earlier in the video. So we have just shown that applying the overall effect of BC and then A is the same as applying C, then B, then A. Similarly, (AB)C means apply C then the overall effect of AB. But applying overall effect of AB is equivalent to applying B then A. So we can just apply C then B then A and get the same thing. Since both A(BC) and (AB)C decompose to applying C then B then A, we have that A(BC) = (AB)C

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[fluxtwee]

I love the small details like how Composition was colored to look like it is a composition of a rotation and a shear.

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[sinaazartash]

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[forthrightgambitia]

7:20 Also, having the intuitive understanding of it means that when you get outside of high school or undergrad you can actually use it to solve new problems rather than answer exam questions as it isn't just an algorithm but a way of thinking about the relationships between sets of dimensions. This is really useful if you are into data science.

[KieranBorovac]

Something interesting: I watched this series before learning about matrices in school, and it was extremely helpful to have this conceptual grounding.

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Sir you have no match. I have many books of mathematics none of them explains the basic concepts.They just explain both basic and advanced concept in a way that we would memorize them, without understanding the essence.You make mathematics real and alive and make us get the real feeling of it. Keep it up and thanks

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[IanBlood]

I spent a few minutes being confused about the associative property of matrix multiplication, but I think the key is to remember that matrices are really transformations, which are really functions, and when we multiply matrices we are really *composing functions*. So, ABC can be thought of as the composition a(b(c(x))). Now we can see that if we were to define some other function, q, as the composition of a and b, i.e., q(x) = a(b(x)), then a(b(c(x))) = q(c(x)). Likewise, we could define a function z that is the composition of b and c, i.e., z(x) = b(c(x)), so a(b(c(x))) = a(z(x)). So, q(c(x)) = a(z(x)), and this is pretty much the same as saying (AB)C = A(BC), I think... Am I right?

[ambientscience]

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[jeromej.]

Good lords! I've never been thought what matrices represented... this changes everyhing! and makes so much more sense! thanks!!

[alfredwong]

"Good explanation > Symbolic proof "
Exactly. I've been screaming this in my mind every time I see math. If my middle school teachers taught like this I wouldn't have hated math.
Thank you so much :D

[emvv]

“Reading from right to left is strange”

Being a weeb is paying off

[Sentas]

Our professor at our university in Germany suggested your YouTube channel to us because he couldn't properly represent the 3 dimensions on the board, and it has been very helpful to me. Thank you for your videos.

[amirwagih]

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