Video: Determining Linear Transformation Using Matrix Representation

Video Transcript

Consider the linear transformation which maps one, one to three, seven and two, zero to two, six. Find the matrix 𝐴 which represents this transformation. The answers to choose from are A) 𝐴 equals one, two, three, four, B) 𝐴 equals three, two, four, one, C) 𝐴 equals two, three, one, four, or D) 𝐴 equals three, one, four, three.

There is also a second part to this question which we’ll come onto. So, when we’re looking to do a linear transformation and we’re looking at matrices, we can say that a linear transformation that transforms a point 𝑝 which is 𝑥, 𝑦 into 𝑝 prime 𝑥 prime, 𝑦 prime can be written as 𝑥 prime, 𝑦 prime is equal to the matrix 𝑎, 𝑏, 𝑐, 𝑑 multiplied by the matrix 𝑥, 𝑦. Where 𝑎, 𝑏, 𝑐, 𝑑, so this matrix, is the transformation matrix.

So therefore, we can say that 𝑎, 𝑏, 𝑐, 𝑑, so our transformation matrix, multiplied by the matrix one, one, which we get from our point one, one, is equal to the matrix three, seven. So therefore, from this, we can form a couple of equations.

First of all, we’ve got 𝑎 multiplied by one which would just give us 𝑎. And then, 𝑏 multiplied by one because it’s gonna be 𝑏 multiplied by the bottom one in our matrix. So, this is gonna give us 𝑏. And then, this is gonna be equal to our 𝑥-coordinate, or our top number in our matrix which is three. So, we can say that 𝑎 plus 𝑏 is going to be equal to three. And I’m gonna call that equation one.

So, now in the same way, we could look at the bottom row. So, 𝑐 multiplied by one gives us 𝑐. And 𝑑 multiplied by one gives us 𝑑. And this is gonna be equal to the 𝑦-coordinate, or the bottom number on our matrix, which is seven. So, we got 𝑐 plus 𝑑 is equal to seven. And this is gonna be equation two.

So, now we can use the same method on our second points. So, we’ve got 𝑎, 𝑏, 𝑐, 𝑑 the matrix, which is our transformation matrix, multiplied by matrix two, zero is equal to the matrix two, six. And we got those from our points. So therefore, this is gonna give us two 𝑎, well, then, it’d be add zero, and that’s because we’ve got 𝑏 multiplied by zero, so we’re not gonna put this in, is equal to two. So therefore, we’ve got two 𝑎 is equal to two. And that’s equation three.

And then, for equation four, we get two 𝑐 is equal to six. And that’s because we had 𝑐 multiplied by two, which is two 𝑐, 𝑑 multiplied by zero, so just zero, and then, like I said is equal to six, which is the 𝑦-coordinate, or the bottom number of our matrix on this second point, which is our image. So, now what we need to do is solve our equations to find 𝑎, 𝑏, 𝑐, and 𝑑, so our transformation matrix.

Well, first of all, we’re gonna solve three because that’s the easiest one to solve. So, we’ve got two 𝑎 is equal to two. So therefore, if we divide both sides of the equation by two, we get 𝑎 is equal to one. So, next we solve equation four. Well, equation four is two 𝑐 is equal to six. So again, if we divide both sides of the equation by two, we’re gonna get 𝑐 is equal to three. So, we’ve now found 𝑎 and 𝑐.

So, next, we’re gonna use the value of 𝑎 that we found which was 𝑎 equals one. And we’re gonna substitute it into equation one to find out what 𝑏 is. So, we get one plus 𝑏 is equal to three. So therefore, 𝑏 is equal to two because it’s three minus one. And then, finally, we’re gonna substitute 𝑐 equals three into equation two. So, we get three plus 𝑑 is equal to seven. So therefore, 𝑑 is gonna be equal to four. So therefore, we can say that our transformation matrix is gonna be one, two, three, four.

So therefore, we’ve circled the correct answer which is A. And that’s 𝐴 which is our transformation matrix is equal to the matrix one, two, three, four. Okay, great, so now we can move on to the second part of the question.

So, the second part of this question asks, where does this transformation map one, zero and zero, one?

So, first of all, let’s start with the point one, zero. So, we’re gonna multiply this by our transformation matrix. So, we’ve got the matrix one, two, three, four multiplied by, and we’ve put it in matrix form, so one, zero. So then, to find out what our 𝑥𝑦-coordinate’s going to be, we multiply one by one, cause that’s the corresponding first term from the first row by the corresponding first term from the first column, then plus two multiplied by zero. And then, we’ve got three multiplied by one plus four multiplied by zero, which is gonna give us the matrix one, three. So therefore, the image of one, zero is going to be one, three.

So, now we’re gonna move on to the point zero, one. So again, we’re gonna multiply zero, one by the transformation matrix, which is one, two, three, four, which is gonna give us one multiplied by zero plus two multiplied by one and three multiplied by zero plus four multiplied by one, which will give the matrix two, four. So therefore, the image of zero, one is going to be two, four. So, what we can say is the transformation maps one, zero and zero, one to the points one, three and two, four, respectively.