### Video Transcript

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

There is also a second part to this question which weβll come onto. So, when weβre looking to do a linear transformation and weβre looking at matrices, we can say that a linear transformation that transforms a point π which is π₯, π¦ into π prime π₯ prime, π¦ prime can be written as π₯ prime, π¦ prime is equal to the matrix π, π, π, π multiplied by the matrix π₯, π¦. Where π, π, π, π, so this matrix, is the transformation matrix.

So therefore, we can say that π, π, π, π, 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, weβve got π multiplied by one which would just give us π. And then, π multiplied by one because itβs gonna be π multiplied by the bottom one in our matrix. So, this is gonna give us π. And then, this is gonna be equal to our π₯-coordinate, or our top number in our matrix which is three. So, we can say that π plus π is going to be equal to three. And Iβm gonna call that equation one.

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

So, now we can use the same method on our second points. So, weβve got π, π, π, π 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 π, well, then, itβd be add zero, and thatβs because weβve got π multiplied by zero, so weβre not gonna put this in, is equal to two. So therefore, weβve got two π is equal to two. And thatβs equation three.

And then, for equation four, we get two π is equal to six. And thatβs because we had π multiplied by two, which is two π, π multiplied by zero, so just zero, and then, like I said is equal to six, which is the π¦-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 π, π, π, and π, so our transformation matrix.

Well, first of all, weβre gonna solve three because thatβs the easiest one to solve. So, weβve got two π is equal to two. So therefore, if we divide both sides of the equation by two, we get π is equal to one. So, next we solve equation four. Well, equation four is two π is equal to six. So again, if we divide both sides of the equation by two, weβre gonna get π is equal to three. So, weβve now found π and π.

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

So therefore, weβve circled the correct answer which is A. And thatβs π΄ 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, letβs start with the point one, zero. So, weβre gonna multiply this by our transformation matrix. So, weβve got the matrix one, two, three, four multiplied by, and weβve put it in matrix form, so one, zero. So then, to find out what our π₯π¦-coordinateβs going to be, we multiply one by one, cause thatβs 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, weβve 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 weβre gonna move on to the point zero, one. So again, weβre 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.