 Another important feature of a linear transformation is called its range. And this has the same meaning that it has when dealing with functions. Given any linear transformation that sends one vector v to another vector u, the range consists of all possible output vectors u. And given this definition, there's a natural problem that arises. Suppose I have a linear transformation. How do I find the range of that transformation? So let's see if we can solve this problem. So first of all we might want to remember that any linear transformation can be described in terms of a system of linear formulas, and the coefficients of these formulas can be turned into our transformation matrix. So any linear transformation has some corresponding transformation matrix associated with it. And remember that this transformation matrix is going to be the same as the coefficients of the system of linear equations that describe how to obtain the vector u from the vector v. Since we want to find the range of the linear transformation, let's remember the definition. It's going to consist of all possible output vectors u. Well that seems like a really daunting task. How do I find every possible output vector? And the answer to that is, I don't know. But I can probably determine if a particular vector is in the range. And that's because if I know the components of the vector, then they're going to be the constants in a system of linear equations. And this leads to an important strategy for solving problems in linear algebra. If you want to solve a problem, set up and solve a system of linear equations. So to find the range, we'll set up an equation. So if I have a vector, it will be in the range if the system of equations is solvable. And I can answer this by row reducing the augmented coefficient matrix. Well, paper is cheap, pixels are cheaper. Let's try and solve this and see where we go. So suppose we want to find the range of our transformation. We can row reduce the augmented coefficient matrix. So the first row has pivot two. We'll multiply the second and third rows by two. We'll add multiples of the first row to clear out the entries below the first row pivot and get a new matrix. Moving on to the second row, the pivot is seven. So we'll multiply the third row by seven and add a multiple of that second row to eliminate the entry below the second row pivot. Since v4 is never the leading variable, it'll be our parameter. And the thing to notice here is that whatever the values we have for u1, u2, and u3, we will be able to find a solution in terms of v1, v2, v3, and v4. So since this transformation takes vectors with four components to vectors with three components, what this means is that any vector with three components can be produced by transforming a vector with four components. So the range includes any value of those vectors with three components.