 So let's take a look at the range of this transformation. Since this is a 4 by 2 matrix, we know that T takes vectors in R2 and sends them to vectors in R4. So we might say that T applied to vector v1, v2 gives us vector u1, u2, u3, u4. So we want to find the range, so we want to find what u1, u2, u3, 4 could be. So we can find this by solving our system. So if we row reduce our augmented coefficient matrix, and notice we end up with two rows of zeros. So if there's any hope of this system being solvable, we'll need both of these expressions to be equal to zero. And so our range is going to require that u1, u2, u3, u4 satisfy this system of linear equations. And so this gives us a new homogeneous system, which we can row reduce, and we can parameterize our solutions. And so our range will be the linear combinations of the vectors.