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let's say that we've got a position vector P and it is equal to we'll represent it as a column vector right over here to 1 so if we wanted to plot this and that's what I will do let's plot it so if we were to plot it that's my y-axis this right over here is my x-axis x-axis so if we assume that the first entry here is the x coordinate 1 2 comma 1 so our position vector is going to be right over here we could represent it we could represent it in this kind of vector symbol right like that where the tail is at the origin and it's the tip the head is at that point or we could say it's really representing this position right over here on the coordinate plane now what I want to do in this video is apply a transformation to this position vector and the way I'm going to do it is I'm going to multiply our vector P by a matrix and then the resulting product is going to give me another position vector so what do I mean by that well I could have a transformation matrix capital T and let's just say it is equal to 2 let's say it's equal to 2 1 negative 1 & 2 so what happens if I multiply 2 times P let's do that right over here T T times P so first let's just make sure that this that that this is actually a valid operation that that matrix multiplication or matrix vector multiplication is defined here so let's just look at what T and P look like we already know what they look like so I'm going to copy and paste copy and paste that is T and this is P right over here so copy and paste and let's see can we multiply can we multiply a 2 by 2 matrix times a column vector like this which is essentially a 2 by 1 matrix well sure as we know matrix multiplication is only defined or at least conventional matrix multiplication is only defined if the first matrix the first matrices number of columns number of columns is equal to the number of rows in the second Matrix right over here we see there both of those are two so this is going to result in a two by one matrix so this is going to result in a two by one matrix now what's interesting about that what's interesting about that is that this is another column vector this is another position vector so you took this vector P multiplied it by this transformation vector and we're going to get another two by one which we could think of as a position vector which we could plot so what's essentially happened is is this this transformation vector has moved has essentially given this point it has given us a new point so let's think about what that is so this first entry here we're essentially thinking about we're going to deal with the first row and the first and the only column really so let me do it use a color that I haven't used yet so we're going to think about this row in this column so is going to be 2 times 2 which is 4 plus 1 times 1 which is 1 or this is just going to be equal to 5 and then for the second entry right over here it's going to for the second row first column it's going to deal with the second row here and the one-and-only column right over here so negative 1 times 2 is negative 2 plus 2 times 1 which is 2 negative 2 plus 2 is just equal to is just equal to 0 so now we have the position 5 comma 0 which is right over here so 1 2 3 1 2 3 4 5 so we started with this point this this position vector I guess you could say P has transformed it to this other position vector and I guess we could call this I want we call this P P Prime so that if you wanted to draw them as vectors this right over our kind of in traditional vector form that right over there is P Prime this right over here is P this right over here is P this is P actually make sure there's a prime and this is P Prime and the way we got from P the P Prime is using this transformation matrix