Visual representation of transformation from matrix

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 so they don't give us any any coordinates a for the vertices of the quadrilateral which are really the most useful points to use when thinking thinking about potential transformations so just 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 and I encourage you to do it on your own first pause the video come up with some coordinates for this for this for this red quadrilateral and then see what transformation you get in which of these seem to be closest to the one that that you got so I'm assuming you've had a go at it and 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 that is let's say this is a square and so this is one comma one so I'll just write that as the column vector as a column vector 1 1 and let's say this one then would be this is this one right over here would be 1 negative 1 1 negative 1 and then this one over here would be this point right over here could be represented by the position vector negative 1 negative 1 negative 1 negative 1 negative 1 negative 1 and then finally finally this point right over here could be represented by the position vector negative 1 1 negative 1 1 so let's see what the transformation matrix would do when it transforms these four points and the way I'm going to think about it let me just take our transformation matrix so 3 0 0 3 and I'm going to multiply it by a 2 by 4 matrix that multiple that represents all of these position vectors so I'm going to multiply it by so we have this point 1 1 that's our first point we have the point 1 negative 1 1 negative one we have this point which is negative 1 1 negative 1 1 and then we have this point which is negative 1 negative 1 negative 1 negative 1 and that these were convenient points to pick since it didn't give us the points because it'll make the math fairly straightforward so what is this going to be equal to we have a 2 by 2 2 by 2 times a 2 by 4 matrix multiplication is defined here because we have the same number of columns as we have rows right over here and we're going this is going to result in a 2 by 4 matrix so this is going to give us another 2 by 4 matrix which makes sense because we're going to need 4 column vectors here for our four new transformed points and let's figure out what it is so the first the first column vector right over here we could think about this row in this column so or exit the first position right over here this first row first entries this row this column the second one's going to be this row the second row in the first column so let's see 3 times 1 plus 0 times 1 well that's 3 plus 0 so this is going to be 3 and then over here 0 times 1 plus 3 times 1 is going to give us 3 as well I think you see already a pattern here is that when you multiple I guess you could say the x coordinate for each of these vectors we're involving this row and this row we really just multiply it we just we really just multiply the 3 times the x coordinate here and then we don't involve the y coordinate because we're multiplying it times 0 we'll see that over and over again so over here you have 3 times 1 3 times 1 plus 0 times negative 1 so it's just really 3 times 1 which is 3 and then over here you see this for the y coordinate each time we're only involving the y coordinate of the of the of the point before transformation you see 0 times 1 is 0 so we're essentially not thinking about the x coordinate and then it's just 3 times negative 1 which is negative 3 so you see what it's just doing is it's just it's just scaling each of these up by a factor of 3 or three times zero times three times negative one is negative three plus zero times one so it's negative three and then it's going to be zero times negative one zero plus three times one is three and then finally 3 times negative 1 is 3 times negative one plus zero times negative one is negative three and then 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 so actually this one seems to be the closest to the one that we're thinking about how do I know that well look at this this this is the point this is the point one comma one right over there so one comma one and it gets mapped to the point 3 comma 3 so 1 2 3 1 2 3 this is 3 comma 3 right over there this got mapped to that and we see it with each of them 1 negative 1 1 negative 1 gets mapped to 3 negative 3 and then we have negative 3 or sorry we have negative 1 1 negative 1 1 negative 1 1 is getting mapped to negative 3 3 and then finally of course negative 1 negative 1 negative 1 negative 1 gets mapped to negative 3 negative 3 so it's definitely the second the second diagram is the one that represents what what hat wasn't that what that represents the transformation matrix T being applied to the red quadrilateral