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Visual representation of transformation from matrix

Video transcript
If the transformation matrix T is equal to three zero zero three, choose which sketch can represent this transformation when applied to the red quadrilateral. This is fascinating! They don't give us any coordinates, save for the vertices of the quadrilateral, which are really the most useful points to use when thinking about potential transformations. So let's just make up some just to see what would happen to the particular coordinates that we're looking at. And I think that will give us enough information to think about this. I encourage you to do it on your own first. Pause the video. Come up with some coordinates for this red quadrilateral. Then, see what transformation you get and which of these seem to be closest to the one that you got. I'm assuming you've had a go at it. Let's just say for the sake of argument that this point right over here. That right there, you could say that's the position vector. I'll represent it as a column vector. Let's say this is a square. So this is one comma one. I'll just write that as a column vector one one. And let's say this one then would be one negative one. One negative one. Then, this one over here could be represented by the position vector negative one negative one. Negative one negative one. Negative one negative one. Finally, this point right over here could be represented by the position vector negative one one. Negative one one. So let's see what the transformation matrix would do when it transforms these four points. The way I'm gonna think about it, let me just take our transformation matrix. So, three zero zero three. And I'm gonna multiply it by a two by four matrix that represents all of these position vectors. I'm gonna multiply it by, we have this point one one. That's our first point. We have the point one negative one. One negative one. We have this point which is negative one one. Negative one one. And then we have this point which negative one negative one. Negative one negative one. These were convenient points to pick since they didn't give us the points because it'll make the math fairly straightforward. What is this going to be equal to? We have a two by two. Two by two. Times a two by four. Matrix multiplication is defined here cause we have the same number of columns as we have rows, right over here. This is going to result in a two by four matrix. This is going to give us another two by four matrix, which makes sense, cause we're going to need four column vectors here for our four new transformed points. Let's figure out what it is. The first column vector, right over here. We can think about this row and this column. Or actually, the first position right over here the first row, first entry is this row, this column. The second one's going to be the second row and the first column. So let's see, three times one plus zero times one. Well that's three plus zero so this is going to be three. Then over here, zero times one plus three times one is going to give us three as well. I think you see already a pattern here. To get the the x-coordinate for each of these vectors, we're involving this row. And this row, we really just multiplied the three times the x-coordinate here and then we don't involve the y-coordinate because we're multiplying it times zero. We'll see that over and over again. So over here you have three times one. Three times one plus zero times negative one. So it's just really three times one, which is three. Then over here you see this for the y-coordinate each time we're only involving the y-coordinate of the point before transformation. You see zero times one is zero. So we're essentially not thinking about the x-coordinate. And then it's just three times negative one, which is negative three. You see what its doing, it's just scaling each of these up by a factor of three. Three times negative one is negative three plus zero times one, so it's negative three. Then, it's going to be zero times negative one is zero plus three times one is three and then finally, three times negative one. Three times negative one plus zero times negative one is negative three. And zero times negative one plus three times negative one is negative three again. So what's going to happen to it? Well, each of these coordinates essentially get pushed out by a factor of three. Actually this one seems to be the closest to the one we're thinking about. How do I know that? Well look at this. This is the point one comma one right over there. One comma one. It gets mapped to the point three comma three. So, one two three. One two three. This is three comma three right over there. This got mapped to that and we see it with each of them. One negative one. One negative one gets mapped to three negative three. Then, we have negative one one. Negative one one. Negative one one is getting mapped to negative three three. Finally, of course, negative one negative one. Negative one negative one gets mapped to negative three negative three. It's definitely the second. The second diagram is the one that represents the transformation matrix T being applied to the red quadrilateral.