Invertible matrices and transformations

- We have two by two matrices here and in other videos we talk about how a two by two matrix can represent a transformation of the coordinate plane of the two dimensional plane where this of course is the x-axis and this of course is the y-axis. What we're doing in this video is visualize these transformations and get a visual understanding for why it's reasonable for A to have an inverse or for why matrix A is invertible and why it's not reasonable for B to have an inverse matrix or why matrix B is not invertible. So just as a reminder transformation these transformation matrices essentially tell us what to do with our unit vectors. For example, we have the one zero unit vector and the first column of each of these matrices tell us what the one zero unit vector the vector that goes one unit in the extraction what it will get transformed to under each of these transformations and then of course you have the unit vector in the Y direction the zero one unit vector and the second columns here tell us how we would transform that. So let's first think about matrix A, matrix A transforms the one zero vector into the two, one vector so the two, one vector is going to look something like that and it transforms the zero one vector into the two, three vector so the two three vector is going to look something like this. So one way to think about it instead of our grid looking like this grid that I already had here which is just our standard coordinate axes. This is defining a new grid that would look like this and I define this new grid by looking at our multiples of what the one zero vector has been transformed into and we also look at multiples of what the zero one vector has been transformed into. So for example, if I were to take this point right here before it is transformed it has one of each of these vectors, well under this first transformation it would be one of each of these vectors. So it's going to be one two one vector plus one two three vector so this point will be mapped to this point and by that same logic this point right over here which is one unit more in the extraction well now it's going to have one unit more in the direction that the X unit vector has been transformed into this point right over here by the same logic will be one unit more in the direction that the Y unit vector has been transformed into so it will be there. And that this point by the same logic will be transformed into that point and so this region that I'm showing you in white will get transformed to this region. Now there's some obvious things going on here. We have a two dimensional area that has been transformed to another two dimensional area and in fact, it looks like it has been scaled up in other videos we have talked about that this scale factor is going to be the absolute value of the determinant of A and it's clear that frankly, not only is this non-zero but it's going to be greater than what it looks like we are scaling up our area but the very fact that this does not equal zero tells us that we were scaling from a two dimensional area to another two dimensional area. And so it's completely reasonable to be able to go back you will, for sure be able to find a transformation that takes you from this region to that region. So that makes us feel good that A inverse is reasonable. Now, as a point of contrast let's think about matrix B right over here. Matrix B transforms the one zero unit vector into the two, one unit vector so it transforms it into this unit vector actually very similarly to how A did it but then let's see what it does to the zero one vector transforms this vector into the four two vector four two is this notice the four two vector is just two of the two one vectors is going in the same direction it just has a different size or different length or a different magnitude. So in this situation is taking things in two dimensional space and it's turning them into combinations of things that sit on this same direction. So everything in two dimensions is going to be mapped to something along this line here. So this region, if I were to apply transformation B it's going to be a region of this line over here anything, if I were to map it using matrix B is going to get mapped onto this line so you're going to something with area and then you're mapping it to something that has no area so the scaling factor here must be zero. So we know since the absolute value of the determinant of B is zero, we could also fairly say that the determinant of B is equal to zero and the how are you going to have an inverse matrix? How are you going to have something that can scale from zero area to something that has area? So we know that B inverse does not exist there's nothing that can transform us back and so there's a couple of things here this reinforces this idea that if your determinant of a transformation matrix is zero you're not going to have an inverse, it's not invertible the other thing to recognize is seeing the patterns in the matrix itself here we saw that the second column is just a multiple of the first column it's twice the first column two times two is four one times two is two you can also view it the other way round the first row is a multiple of the second row and you can play around with the math if you wanted to generalize it and see if that's the case the determinant will always be zero and that's because if you view them as transformations it's going to map to line and you're gonna lose all your area, if you view it as a representation for at least the left-hand side of a system of equations you can think about it as lines that have the same slope but we talk about that in other videos.