Visualizing Composition of Linear Transformations **aka Matrix Multiplication**

Matrix multiplication is an algebraic operation. But we cared about that algebraic operation because it represented a core geometric idea: the composition of linear transformations. In this video I introduce my favorite way to visualizing Linear Transformations in 2D (dynamically!). We then can visualizing the composition and track what happens algebraically at the level of linear combinations or at the level of matrix multiplication.

Typo: The yellow unit vector should be (0,1) not (1,0)

**************************************************
Now it's your turn:
1) Summarize the big idea of this video in your own words
2) Write down anything you are unsure about to think about later
3) What questions for the future do you have? Where are we going with this content?
4) Can you come up with your own sample test problem on this material? Solve it!

Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master math means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK.

***************************************************
►Want more ideas for learning math effectively?
How to Watch Math Videos:

****************************************************
►Want some cool math? Check out my "Cool Math" Series:

****************************************************
Course Playlists:

*****************************************************
*****************************************************

This video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.

BECOME A MEMBER:

MATH BOOKS & MERCH I LOVE:

[hamedhilal]

Your videos are the only ones out there that truly teach us what the heck the computations mean! I owe you so much!

[kathirs]

In the vertical component, it has (1 0) as well. I think this needs to be fixed as (0 1)

[kevinpreston]

@4:54, Algebraic and geometric representations don't agree. If we assume the geometric representation to be correct, then the 'T' transformation maps (0, 1) to (1, -1), not (-1, 1). Anyone else see this discrepancy?

[weisanpang]

Dr Trefor, in your example of T and S matrices, T * S does not equal S * T, contrary to what the last illustration in the video claimed.

[manueljenkin]

Thank you very much. I am now able to write it down easier and visualize it much clearer.

[GadgetGuyU.K.]

This video is amazing! Thanks for posting. Can I ask what software you used for the animation?

[SubtleForces]

To pick up the challenge, though purely conceptually, it seems a combination of a flip and a rotation is a good candidate for a sequence of transformations that is order-sensitive. For example, let's take a 90-degree counterclockwise rotation and a flip along the x-axis. If the sequence is: rotate-flip, the (1 0) vector becomes ( 0 -1). The other way, it becomes (0 1). Obviously, the next question is whether there are classes of transformations that are order sensitive and classes that aren't. But, I guess it is a little too late after a long workday to think about such novel ideas with an old brain. Maybe some general ideas could occur...before the next video sequence. Thanks!

[sureshpatel-sgxw]

sir, I tried to solve the challenge by taking transformation case1 { T then S }and case2 { S then T}. it is strange that on board(animation) we get the same vectors in both cases but on paper, they are different...as TS!=ST here, I got T=([[2, -1], [1, 1]]) and S=([[1, 0], [0, -1]]) (I'm writing rows side-by-side) sir where did i went wrong..??

[asthashrivastava]

Great sir...it is very good explanation nd very useful.

[jacobm]

I would also be very interested to know what software you used for these! Amazing vid!

[anchalgera]

Can you refer to some books to study linear algebra the way you teach, I understand but for revision and solve some questions!

[canyouhandlethetruth]

It's A curved dome not liner and hieroglyphic are covering the inside of it. symbols are making computations constantly and changing constantly. The hieroglyphics are actually making physical changes to the universe to compensate for realiety from one symbol to another. There are slots covering the dome and all of them are changing constantly like computating to keep realiety going. I also heard a voice in a mechanical voice in a unknown language trying to speak.

[Felxaga]

I am very confused... How does the T transform change x1(1, 0) to x1(2, 1)? Same for x2. I believe I understand the S transform however.

[AZskies]

what is the software that you are using that depicts the board?

[nonentity]

I actually find this video more helpful than 3b1b (not to offend him though)

[canyouhandlethetruth]

Its not liner its dome shaped I saw it in real life.