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Current time:0:00Total duration:8:12

Matrices as transformations of the plane

Video transcript

in this video we're going to explore how a two by two matrix can be interpreted as representing a transformation on the coordinate plane so let's just start with some examples or some conceptual ideas so the first conceptual idea is that any point on our coordinate plane here and this of course is our x-axis and this is our y-axis can be represented by a combination of two vectors you could have this vector right over here that goes exactly one unit in the horizontal direction we can represent that as a vector like this one zero when you write a vector vertically like this the convention is that the top number here is what we're doing in the x direction and then the bottom number the zero is what we're doing in the vertical direction or the y direction so this is the one zero vector and then this right over here what would we call that well that we would call the zero one vector because it doesn't go at all in the x direction and it only goes up one in the y direction now just to feel good that any point on the coordinate plane can be represented as a weighted sum of these let's just pick a point at random let's say this point right over here let's call it point a and you could represent that as a vector that looks something like this i'll do it as a dotted line but this could be represented as a positional vector like that and of course if we're thinking about in coordinates we would just say this is at the coordinate 3 comma 1. the x coordinate is a 3 the y coordinate is 1. but if we wanted to express it in terms of a vector we could write it as 3 1. the x direction we're moving 3 from the origin positive 3 in the y direction we're moving 1 to get there and you could see that we can represent this as a weighted sum of these two vectors we can write this as this is the same thing as 3 times our 1 0 vector 1 0 plus 1 times our 0 1 vector 0 1 vector and you could see it visually this yellow vector that points to point a right over there you can have three of this vector one two and then three and then one of the orange vector now i said that i would explain how two by two matrices can represent a transformation and the way that you could think about it is if i have a two by two matrix that looks like this so let me just draw the matrix where the first column is 1 0 and then the second column is 0 1 this just tells you what to do with these two vectors i know this might be a little bit confusing at first but let's let's just walk through it together so the way i've represented it this first column says what is the transformation you want to apply to this one zero vector this first blue vector well we're just keeping it one zero so we're not changing it it's the same one zero vector here one zero vector here likewise zero one vector here zero one vector here so this two by two matrix actually represents what's sometimes known as the identity transformation it maps any point on the coordinate plane back to itself it doesn't change the points but i'm showing this because now i'm going to show a two by two matrix that represents a non-identity transformation for example let me draw the matrix again so let's say i have the matrix instead of 1 0 here i'm going to write a 2 1 here instead of a 0 1 here let me write a 1 2. so in this transformation what we're doing is we're turning this 1 0 vector into a 2 1 vector you're going to see what i'm talking about in a second so what does a 2 1 vector look like let me do it in that color well we go 2 in the x direction 1 in the y direction so it's going to look like this it's going to look like that and then what does a 1 2 vector look like well it goes 1 in the x direction and then 2 in the y direction so it looks like this and the way that this represents a transformation is that anything that was a weighted sum of the 1 0 and the 0 1 vectors originally you can now view as a weighted sum of the two one and the one two vectors and so we can now think of another point a prime that's not going to be three of the one zeros and one of the zero ones we can think of it as let me write it over here as three of the two ones plus one of the one twos one of these one two so where would that put that now well we're going to do three of the two ones so this is one of them this is two of them and then this is three of them and then i'm going to have one of the one two vectors this orange vector right over here and so i'm going to have one of those and so that will take me to a prime this is my new point after the transformation i go to a prime so i've gone from this point a to this point a prime and this two by two matrix is telling us how to transform now we can do that with multiple points now if i talk about a point here at the origin that was originally at the origin point b well that's zero of the orange vector zero of the blue vectors so even after the transformation is going to be zero of the two one vector and 0 of the 1 2 vectors so it's just going to stay in place it's just going to map to itself so b is equal to b prime and we could also imagine another point let's say right over here let's call that point c well point c is originally two of the blue vectors and none of the orange vectors so after the mapping it'll be two of the two one vectors and none of the one two vectors so two of the two ones so if you go one two and then none of the one twos you're going to get c prime right over there and so notice if originally you had a triangle between a b and c let me draw it like this so originally you had this triangle abc what is it now gotten mat what is it now mapped to well it's now mapped to this big triangle do my best to draw it relatively straight so we land on target so that's that side and then we have this side going from b prime to c prime and then we want to connect that side from c prime to a prime now you might be saying so how do you know that the lines map onto other lines how do you know this transformation didn't all of a sudden make this line squiggly or zigzag and that's one of the interesting properties of the type of transformation we're talking about a two by two matrix will represent a linear transformation and there's two ways to think about it in this context a linear transformation will always map the origin onto itself and it will always map a line onto another line it won't turn that line into a curve or it won't make it zigzag somehow now the last thing you might be wondering is hey what about all these transformations we had from geometry these similarity transformations things like rotations reflections dilations can you do those with matrices and the simple answer is yes you can do them as long as you keep the origin in place and you can actually using a 2x2 matrix come up with a whole series of other linear transformations that are much more let's call it exotic than just the rotations reflections and dilations