yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Composing 3x3 matrices | Matrices | Precalculus | Khan Academy


2m read
·Nov 10, 2024

So, we have two three by three matrices here: matrix A and matrix B. We could, of course, view each of them as a transformation in three-dimensional space.

Now, what we're going to think about in this video is the composition of A and B. So, you could think of this as the transformation where you apply B first, and then you apply A after that. Then, we can represent that by another 3x3 matrix, which is partially completed here. We have the first and the third column here, and so my question to you is: what is this middle column where I have these three blanks? Pause the video and try to work through that.

Alright, now let's work through this together. One way to think about how to construct A of B is that what you're doing is you're taking each of the columns of B and you're thinking about what they would be under the transformation A. So, if you were to apply the transformation A to this column right over here, you would get this column. If you apply the transformation A to this column right over here, you would get this column.

So, what we really need to do is apply the transformation A to this column, to the middle column right over here. And just as a reminder, how this transformation works: a vector (0, 2, 3) you can think of it as 0 of the (1, 0, 0) vector, the unit vector in the x direction, plus two of the (0, 1, 0) vector plus three of the (0, 0, 1) vector.

Now, when you apply the transformation, instead of using these unit vectors, you use the image of them under this transformation. And now, in this situation, instead of a (1, 0, 0) vector, we are going to be using this thing. Instead of a (0, 1, 0), we're going to be using this thing. Instead of a (0, 0, 1), we are going to be using this thing.

So, this middle column, what is transformed by this vector, is going to be zero instead of the (1, 0, 0) one. It's going to be zero of the (-3, -3, 3) vector, and then we have plus two, plus two of the zero let me do that in that purple color, of the (0, -2, 3) vector. Then, last but not least, you're going to have three of the plus three of them—I’ll do that in yellow—the (0, -4, 1) vector.

Now, we just quote do the math. So, when you apply zero times all of this, you're just going to have a (0, 0, 0) vector. So, we can let those all go away, and then you are left with, let’s see, this one is going to be 2 times 0 is 0. 2 times -2 is -4. 2 times 3 is 6. You're going to have that plus 3 times 0 is 0. 3 times -4 is -12. 3 times 1 is 3.

I could have written this a little bit neater but hopefully you get the idea. And then, when we add those two things: 0 plus 0 is 0. -4 plus -12 is -16. 6 plus 3 is 9. And we're done! We have just completed the composition of A of B.

More Articles

View All
Brie Larson Eats a Rhino Beetle | Running Wild With Bear Grylls
So were you kind of adventurous when you were growing up, or– I wanted to be Indiana Jones when I was younger, but then– Really? –I also was super shy. And were you like, sporty? No, not at all. BEAR GRYLLS: So what, more geeky? I was super geeky. …
I’m an IDIOT for getting a credit card...
What’s up you guys, it’s great I’m here. So let’s just say this: I did not expect to get so many views on the unboxing of the JP Morgan Reserve card. So if you’re brand new here, welcome! My name is Graham. I flex credit cards, duplexes, and lotuses. So h…
My Response To Elon Musk (Why Tesla Is Screwed)
What’s up, Elon? It’s Twitter here, and it’s not every day I make a video like this, but we gotta talk about what’s going on with Tesla. Because I have to say, as a two-time Tesla owner, a Tesla stockholder since 2019, and as an investment channel here on…
How To Build The Future: Sam Altman
We said from the very beginning we were going to go after AGI at a time when in the field you weren’t allowed to say that because that just seemed impossibly crazy. I remember a rash of criticism for you guys at that moment. We really wanted to push on th…
Fake Beams - Smarter Every Day 186
Hey, it’s me Destin. Welcome back to Smarter Every Day! So, if you watch Smarter Every Day for any length of time, you know that it’s about whatever I’m thinking about—like in Eclipse, or how brains work, or helicopters, or management, or whatever. You di…
Back to Camp | Life Below Zero
Sue Aikens has spent the past four months in Fairbanks recovering from surgery. As Kavik River Camp’s sole proprietor, her absence has left this site uninhabited and unprotected. “I have to visually check everything, behind any corner, any trailer. It’s …