Now, what's the point of telling about all these different geometric transformations? Well, if you want to do any shape alteration, save all the pixels in an image of a face or something like that, then you can always make that shape change out of some combination of rotations, shear structures and inverses. That is if I first apply one translation A1 to a vector r, then that makes some first change. Then if I apply another translation A2 to the results of that, then I can perform a composition of the two translations. What I've done is I've performed first A1 and then A2 to r. Now, maybe this isn't so obvious, so let's slow down into a concrete example. Let's start out with our basis vectors. So we've got e1 as being one, zero, and e2 as being zero, one. Now, let's take our first transformation A1 as being a 90 degree anticlockwise rotation. So what happens if we rotate this by 90 degrees anticlockwise? So e1 comes down to some transformed e1, let's call it e1 prime of zero, minus one. So put zero, minus one in there and our e2 rotates down here, to be some transform version of e2, which will then be e2 primed, which will be one, zero. So then we've got an overall A1 which describes a 90 degree rotation. Now, let's take another matrix which also transforms our original basis vectors, original e1 and e2. What we'll say is that is a let's say a mirror. So that moves a vertical mirror moves e1 to e1 prime, is going to be minus one, zero. It's going to leave e2, where it was if I just reflect vertically. So my transformation A2 now is going to be minus one, zero, and it leaves the other one the same zero, one. Now, let's ask ourselves what happens if I do A2 to A1. So now I'm going to reflect over the result of doing A1. So e1 prime is actually when I reflect vertically, going to stay in the same place. That's going to give me e1 let's say double-prime for doing it twice. E2 prime it's going to reflect over here, and it's going to become e2 double prime is going to be minus one, zero. So I'm going to have an overall result of doing A2 to A1, which is going to be, well I get e1 prime I first write down zero, minus one and e2 double prime I'm going to write down minus one, zero. Now, I can work out what that is, actually in a matrix way without having to draw it all out by saying, A2 to A1 is doing A2 which is minus one, zero, zero, one, to A1 basis vectors. A1 was zero, minus one, one, zero which is just the two transform basis vectors the single primes. When I do that, I just have to do this matrix to that basis vector and then this matrix to that basis vector. What that looks like, if I do that is I do this matrix to that basis vector so I'll do that row times that column. That gives me minus one times zero plus zero times minus one that gives me zero. That one to that one. So the second row first column zero times zero plus one times minus one gives me minus one. Then I do that A2 translation to the second column. Now, to the second basis vector with A1. So minus one times one, plus zero times zero, of minus one times one second row zero times one plus one times zero gives me zero. So I do the row times the column for all the possible row and column combinations. So that is in fact the same thing. So we can have discovered really how to do matrix composition, or matrix multiplication doing one matrix to another transformation matrix. Notice that geometrically, we can show that A2 then A1, isn't the same as doing the translations in the other order first A1 and A2. So just watch that for a minute. So this one we've done A2, and then what if we then do A1. A1 is a 90 degree rotation so if we rotate e1 prime, then we find that e1 double prime, would be equal to our original e2 that's A1. Our e2 primed, which was just staying where it was, when we rotate that down, then that'll come down to here. That'll come down to one, zero, so our e2 double prime will be one, zero. Let's look how matrix wise. So if we do A1 to A2, A1 was zero, one, minus one, zero. If we do that to A2, which was minus one, zero, zero, one that we're going to do that matrix multiplication. What that's going to give us is zero times minus one plus one times zero, zero, then this one times this one, minus one times minus one plus zero times zero is one. This row second column zero, zero, one, one and second row second column minus one, naught, naught, one gives me naught. So that is the first column is zero, one, and the second column is one, zero. So that isn't the same as doing the operations in the other sequence. You see these minus signs are flipped. In fact what happened is, this composition rotating and then flipping over, is the equivalent of reflecting the whole lot, the original basis vectors in this mirror, and flipping and then rotating round it effect is I've just flipped, my original axis e1 and e2 which is putting a mirror in here. So doing the two operations in opposites in different sequences, doesn't give you the same operations. What we've shown is that matrix multiplication isn't commutative. A2, A1 isn't the same as A1, A2. So we have to be very careful with matrix multiplication. We can do them in any order meaning they're associative. That is if we do A3 to A2, to A1, we can do A2, A1 and then do A3, or we can do A3, A2 to A1. So we could do that, and then that. Those are the same, they're associative. But we can't interchange the order. We can't swap them around. It's not commutative. So this is interesting. There seems to be some deep connection between simultaneous equations, these things called matrices, and the vectors we're talking about in the last module. It turns out the key to solving simultaneous equation problems, is appreciating how vectors are transformed by matrices. Which is the heart of linear algebra.
