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




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Multiplying two matrices represents applying one transformation after another. Many facts about matrix multiplication become much clearer once you digest this fact.

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Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced.

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  • This series is simply awesome. I got engineering master, and working in quantitative finance now. Lots of matrices during my study and work. However, I never understood how people came up with this kind of method. When I was in college, I had no idea why we need to learn this. After this series, every dot are finally connected. Thank you, and, hats off to all those great great mathematicians. What we are enjoying now are all based on their great and genius work hundreds of years ago.


  • I don't who you are but I must say you are the "Einstien" modern era.


  • Sorry I don't really get the part on associativity. Why is it C->B->A for both examples? Doesn't the parenthesis for (AB)C mean that you should do B->A first then -> C?


  • Guy, you're incredible!


  • I can't be more thankful for the creators of this series. I am truly grateful.


  • hermano te amo, gracias por existir.


  • There are few who still misunderstood like me as follows: 1. Why commutativity is not satisfied, where as Associativity does satisifed by Matrix Multiplication. Just to make things clear, still ABC!=CBA. Its just A(BC)=(AB)C. In other words, on both sides of associativity, 1st transformation is C, 2nd Transformation is B and 3rd transformation is A.


  • Only in 2020, a brilliant guy who understands this shit is putting efforts to make it so lucid to everyone. I feel fortunate. The creator himself possible cannot imagine what kind of butterfly effect these teaching may lead to. Thank you so much.


  • I've only seen the first 4 videos in the series, and I've gained more valuable intuition than my semester long engineering linear algebra course. Thank you!


  • What you just taught here, it just blew me away. Never had I ever given thought to liner algebra in such a light. I am glad that i found this channel.


  • One suggestion, Please decrease your speed of explaining the concepts. It is a bit faster.


  • you are amazing


  • can you change the school system


  • Wow


  • This is the exact video I was looking for.


  • Linear Algebra review day2


  • Thanks bro, you are legend. Very helpful.


  • @3Blue1Brown: Grant, can we geometrically think of the special condition when matrix multiplication will be commutative? That is to say, under what type of transformations will the order of applying them not matter?


  • I didn't like the associativity proof. (AB)C is not applying C, then B and then A as written, despot being equivalent to such, but you stated it as if it was already proven to prove it. As written, it is the transformation C, and then the transformation that is (AB)
    Oh I get it now


  • Multiplying two matrices has the same geometric meaning as applying one transformation then another.


  • Not quite sure on understanding i head and j head movement in the shear matrix. I see that it has to have does numbers in the shear matrix , but i don't understand how it is cooked up.