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

Using matrices to transform the plane: Mapping a vector | Matrices | Precalculus | Khan Academy


3m read
·Nov 10, 2024

Let's say that we have the vector (3, 2). We know that we can express this as a weighted sum of the unit vectors in two dimensions, or we could view it as a linear combination. You could view this as (3) times the unit vector in the (x) direction, which is ((1, 0)), plus (2) times the unit vector in the (y) direction, which is ((0, 1)).

We can graph ((3, 2)) by saying, okay, we have three unit vectors in the (x) direction. This would be one right over there, that would be two, and then that would be three. Then we have plus two unit vectors in the (y) direction, so one and then two. We know where our vector is or what it would look like. The vector ((3, 2)) would look like this.

Now, let's apply a transformation to this vector. Let's say we have the transformation matrix. I'll write it this way: (\begin{pmatrix} 2 & 1 \ 2 & 3 \end{pmatrix}).

Now, we've thought about this before. One way of thinking about a transformation matrix is it gives you the image of the unit vectors. Instead of being this linear combination of the unit vectors, it's going to be this linear combination of the images of the unit vectors when we take the transformation. What do I mean? Well, instead of having (3(1, 0)), we are now going to have (3(2, 1)). Instead of having (2(0, 1)), we're now going to have (2(2, 3)).

So I could write it this way. Let me write it this way: the image of our original vector, I'll put a prime here to say we're talking about its image, is going to be (3) times instead of ((1, 0)), it's going to be times ((2, 1)). That's the image of the ((1, 0)) unit vector under this transformation. Then, we're gonna say plus (2) instead of ((0, 1)). We're gonna look at the image under the transformation of the ((0, 1)) vector, which the transformation matrix gives us, and that is the ((2, 3)) vector.

We can graph this. If we have ((3, 2)) and ((2, 2)), what I could do is overlay this extra grid to help us. So this is ((2, 1)), that's ((1, 2)). ((1, 2)) is ((2, 2)).

So, we have ((3, 2)) right over here. Let me do this in this color. This part right over here is going to be this vector. The ((3, 2)) is going to look like that. Then to that, we add ((2, 3)). So this is going to be (1, 2), and then (3). So this is going to be (1, 2, 3) and then we have ((2, 2)). So we end up right over there.

Let me actually get rid of this grid so we can see things a little bit more clearly. Here we have in purple our original ((3, 2)) vector. Now the image is going to be ((3, 2)) plus ((2, 3)).

So the image of our ((3, 2)) vector under this transformation is going to be the vector that I'm drawing right here. When I eyeball it, it looks like it is the ((10, 9)) vector. We can verify that by doing the math right over here.

So let's do that. This is going to be equal to (3 \times 2 = 6), (3 \times 1 = 3), and we're going to add that to (2 \times 2 = 4), (2 \times 3 = 6). Indeed, you add the corresponding entries: (6 + 4 = 10), and (3 + 6 = 9), and we're done.

The important takeaway here is that any vector can be represented as a linear combination of the unit vectors. Now, when we take the transformation, it's now going to be a linear combination not of the unit vectors, but of the images of the unit vectors. We saw that visually, and we verified that mathematically.

More Articles

View All
Managing the New York City Power Grid | Breakthrough
I never thought that something as simple as like an LED light bulb could make such a big impact. There was one project that I worked on where the team that I was working with thought, “Well, what about if we just screwed a bunch of LED light bulbs all thr…
Startup Investor School Day 1 Live Stream
And the way the course is organized is there’s a lecture and then there’s a Q&A afterwards. So please hold your questions until the Q&A session at the end unless an instructor explicitly says they want questions during their talk. I will also take…
How To Become The World's First Trillionaire
Everyone is looking to make a quick buck. Whether it be a group of kids running a lemonade stand or a multi-billion dollar company making new cutting-edge technology, everyone wants to be rich. To be among the ranks of Bill Gates, Warren Buffett, Mark Zuc…
Character change | Reading | Khan Academy
Hello readers! One of the wonderful things about stories when they’re given the room to grow and expand is the idea of character change or growth over time. Characters in stories are just like real people; they have the capacity to change, to make mistake…
Is Time Travel Possible?: Mind Blow (Special Edition)
John tier claims to be a time traveler from the year 2036, and his time machine kind of looks like a certain DeLorean whose parts you can actually buy. Vauce Kevin here; this is Mind Blow: Time Travel Edition. This is a real time machine, sort of. Dr. Ron…
EVERYTHING WRONG With My Tesla Model 3 After One Year
What’s up you guys? It’s Graham here. So, as some of you know, last year I bought a Tesla Model 3. In typical me fashion, I tried to be as frugal as possible to get the car’s price down as much as I could by skimping on all of the options. I decided to p…