 Now let's try and find the transformation matrix for a linear transformation. Since T takes vectors with three components and produces vectors with two components, then the transformation matrix T must be a 2 by 3 matrix and corresponds to the formulas shown. Remember the transformation matrix will just be the coefficients of these formulas. And since we don't know what these coefficients are, we can try and set up a system of equations that will allow us to find them. Since our transformation applied to the vector 1 for negative 1 gives us the vector 1, 1, we could substitute in these values for v1, v2, v3, u1, and u2, which will give us two equations in our system. And now we can try and solve the problem. But before we do that, we might want to collect a little more information, despite what you may have been told facts are actually useful and you may want to base decisions on them. And part of that process is not just stopping with one fact, but looking at what else we have. And in this case we know what the transformation does to the vector 2, 0, 3. It gives us the vector 0, 3, and so this gives us another set of equations. And at this point, since we've made use of the fact that the transformation applied to u and v give us certain vectors, we should feel free to ignore any inconvenient truths like what the transformation does to w. Of course, if we do that, we're likely to end up with a wrong answer with all of the consequences that that entails. So let's make use of this third fact that the transformation applied to w gives us the vector 1, 5. And this will give us another set of equations. And this analysis gives us 6 equations with 6 variables, t11 through t23. And so I can put them all together into a single set of 6 equations with 6 unknowns. And we can express this system using a 6 by 6 coefficient matrix, augmented by a column of constants. Row reducing this matrix will give us the entries in our transformation matrix. While I could solve this problem by row reducing this 6 by 6 coefficient matrix, a good habit to get into as a mathematician and as a human being is to ask yourself the question, while I can solve the problem this one way, is there a second way that we can solve the problem? And once you've done that, you might ask yourself, is there a third way or maybe a fourth way? As an alternative to row reducing the augmented coefficient matrix, we note that the formulas will give us the entries t11 and t21 if v1 is 1 and v2 and v3 are both equal to 0. And this corresponds to the problem of finding the transformation applied to the vector 1, 0, 0. Now since t is a linear transformation and we know what it does to u, v, and w, we also know what it does to any linear combination of these vectors. And so that means we want to try and find 1, 0, 0 as a linear combination of u, v, and w. So we want to express 1, 0, 0 as a linear combination of u, v, and w, which will give us this vector equation, and we can reduce the augmented coefficient matrix, and that gives us our solutions. And so we can express 1, 0, 0 as a linear combination of u, v, and w. So because t is a linear transformation, and I now know what 1, 0, 0 is as a linear combination of the vectors u, v, and w, I can find what t of 1, 0, 0 is going to be. So first I'll express 1, 0, 0 as our linear combination of the vectors u, v, and w, and then use the properties of a linear transformation to evaluate what that transformation result is going to be. And so the components of this vector are going to correspond to the entries of the first column of our transformation matrix. By the same reasoning, the transformation applied to the vector 0, 1, 0 will give us the entries in the second column of our transformation matrix. So we'll express 0, 1, 0 as a linear combination of our vectors u, v, and w, which allows us to determine what the transformation does to this vector, and we get our second column in our transformation matrix. And likewise the transformation applied to 0, 0, 1 will give us the entries in the third column of our transformation matrix, and so we could find the value of this transformation in the same way. And if you're incapable of independent thought, for example if you're a robot, computer program, or politician, then you just do the same thing over and over again without any hope of ever progressing. On the other hand, if you are capable of independent thought, then one possibility is to use other ideas to make the process more efficient. And in this case, if we go back to our linear formulas, remember that these gave us the transformation vectors. And since we know what the transformation does to a couple of vectors, and we've determined some of the coefficients tij, then we can use this information to find the last pair of coefficients. So for example, we know the transformation applied to 1 for negative 1 gives us the vector 1, 1, and we know the four coefficients t11 through t22. So we can substitute in our values and get an equation in t13 and in t23, which we can solve to find those last two coefficients and our transformation matrix.