 So let's take a further look at linear transformations and how they fit into linear algebra. So, for example, let's say we have this transformation matrix that goes from some set of vectors to another set of vectors, and we want to do several things. Let's first of all figure out where these vectors live. Next, we want to, if it's even possible, to see what our transformation does to the vector 215, to the vector 1 for negative 1, 0, and to the vector 1, 3. So remember our transformation matrix corresponds to the coefficients of the formulas for the vector components. So we can use the transformation matrix to write our formulas. So the first row of the matrix corresponds to the first formula, 3v1 plus 1v2 plus 1v3 equals u1. Our second row of the matrix corresponds to our second formula, 2v1 plus 4v2 plus 7v3 equals u2. Our third row corresponds to the formula minus 4v1 minus 1v2 plus 1v3 equals u3. And finally the fourth row corresponds to the formula 1v1 plus 0v2 plus 3v3 equals u4. And that means we're going to take vectors v1, v2, v3, and transform them to vectors u1, u2, u3, u4. And so it appears that this transformation takes vectors in R3 and sends them to vectors in R4. And so we can record that result. Since this transformation takes vectors in R3 and sends them to vectors in R4, then in order to apply the transformation, we must use a vector with three components. So for the vector 215, we have v1 equals 2, v2 equals 1, v3 equals 5. We can substitute these into our formula for u1 and find that 12 is the first component of our vector. Likewise, we can substitute these values for v1, v2, and v3 into our second formula to find the second component of our vector is 43. And we'll write that down. Substituting 215 into our equation for u3 tells us that u3 is equal to negative 4. And finally, u4 is equal to 17. And again, because our transformation takes vectors in R3, then the vector with four components and the vector with two components don't live in the right space to be acted on by our transformation. And so the transformation cannot be applied to them. And so we say that the transformation applied to these vectors is undefined.