Here is another, more expository, way, if this helps:
Choose a basis of W w1, w2, w3, w4. T goes from a space of dimension 6 to a space of dimension 4, so it is surjective. Then we can find v1, v2, v3, v4 such that T(vi) = wi. One can show that those vi's are linearly independent, so they can be extended to a basis by appending u1, u2. If T(ui) isn't zero, it can be represented by some combination of wi's, so ui can be represented by some linear combination of vi's. By contradiction, T(ui) = 0, so these are are the basis of the kernel of T, which is then of dimension 2.