 The real world, being what it is, we can't reasonably expect that every possible linear transformation will have a nice, simple form. But luckily enough, we're not working in the real world, we're working in the mathematical world. And, while this may seem to limit the utility of mathematics, it is a remarkable fact that mathematics works pretty well at describing the real world. There's no reason why this should happen, but because it happens, then it means that we can use mathematics quite effectively. So let's take a transformation from xy to x' y' where x' and y' are defined in terms of nice, simple, linear formulas, which we can describe using a coefficient matrix. Let's prove that this is, in fact, a linear transformation. So part of the reason that we do proof in mathematics is it's a really good way of reviewing some basic ideas. In this particular case, let's review what it means to be a linear transformation. And if we want to prove something as a linear transformation, we have to prove that it meets these two requirements. So first, let's prove that the transformation applied to a sum of vectors is the sum of the transformation applied to the vectors individually. So I'll apply the transformation to the sum of the vectors u and v. So I'll write my vector u as components u1, u2, and my vector v with components v1, v2. I want to apply the transformation to the sum of the two vectors. So I'll add the two vectors using our rules for vector addition. And my transformation says take whatever is in the first component multiplied by a, take the second component multiplied by b, and that'll give you the new first component. And likewise, take the first component multiplied by c, plus the second component multiplied by d, and that will give you the new second component. And so I get the transformation applied to the sum of the vectors u plus v. And now let's compare it to the transformation applied to u, plus the transformation applied to v. So if I apply the transformation to u, I get the vector with components a u1 plus b u2, c u1 plus d u2. And if I apply the transformation to v, I get the vector av1 plus bv2, cv1 plus dv2. And these are vectors, so I can add them together. And while we hate to reduce mathematics to bumper stickers, a useful mantra is if in doubt, factor out. And here we see that we can factor a term of a out of both of these, and we can factor a term of b out of both of these, so our first component can be written this way. And likewise, our second component can be written as c times something plus d times something. But this is exactly what we got for the transformation applied to the vector sum u plus v. And so the transformation applied to the vector sum is the sum of the transformation applied to the vectors. So the second part of being a linear transformation is the transformation applied to a scalar multiple has to be the scalar multiple of the transformation. So let's apply our transformation t to a scalar multiple of u, that's k u1 k u2. And applying our transformation to this vector gives us this vector. And again, we'd like this to be k times the transformation applied to u. So we'll apply the transformation to u first, and then scalar multiply the result by k. And we'll compare our two results and we see that they are in fact equal. And so we verified that the transformation applied to a sum is the sum of the transform vectors. And the transformation applied to a scalar multiple is a scalar multiple of the transform vectors. And so that tells us that t is in fact a linear transformation. Can we go the other way? Well, suppose I have a linear transformation, then maybe it's possible to express it as a nice set of formulas. How can we do this? Well, one possibility is the following. If we can actually find these formulas, then we've completed the proof. In other words, if I can find a, b, c, and d that will give us formulas for finding the new x and the new y values, then we will be able to make the claim that any linear transformation can be expressed this way. So let's think about that. One important idea is if I have a linear transformation, then as soon as I know what that linear transformation does to a couple of vectors, I can find what that linear transformation does to every linear combination of those vectors. And so it may be most useful to find what the linear transformation does to the vectors 1, 0, and 0, 1, because then I can express x, y as a linear combination of these two vectors. So suppose my linear transformation applied to 1, 0 gives us a vector a, c. And likewise, our linear combination applied to the vector 0, 1 gives us the vector b, d. Now, it might seem that we're stacking the deck and making sure our linear transformation produces these coefficients a, b, c, and d, but it turns out that what we call these vectors doesn't really make a difference. I could call this vector x1, x2, and this vector x3 and x4, and the only thing that would really change is the names of the terms up here in these formulas. So we'll call this a, c, and b, d with the understanding that it really doesn't make a difference what we call them. So let's see what happens. What does the transformation do to the vector x, y? Well, because it's a linear transformation, I know that this is going to be the transformation applied to x, 0, plus the transformation applied to 0, y. But wait, there's more. Because it's a linear transformation, I know how it handles scalar multiples, and the transformation applied to x, 0 will be the same as x times the transformation applied to 1, 0. And likewise, the transformation applied to 0, y will be the same as y times the transformation applied to 0, 1. But I know what the transformation does to these vectors, and so the transformation applied to x, y is going to be ax plus by cx plus dy. And so now I know that the new x value is going to be ax plus by, and the new y value is going to be cx plus dy, and that's what we wanted to prove. Now, if we put these two problems together, we've proven that any transformation of this form is a linear transformation, and also that any linear transformation must be of this form. And while we only did the proof for vectors with two components, it's easy enough to generalize this, and so these two problems together generalize to vectors with n components, is going to give us our general theorem that if we have a linear transformation, we could express it as a set of formulas. And likewise, any transformation that we can express as a set of linear formulas is going to be a linear transformation, and we call that result a theorem.