The set W below is a subspace of C4. Find the dimension of W

We will appeal to Theorem BS (or you could consider this an appeal to Theorem BCS). Put the three column vectors of this spanning set into a matrix as columns and row-reduce.

The pivot columns are D = {1, 2} so we can “keep” the vectors of W with the same indices and set

and conclude that W = (T) and T is linearly independent. In other words, T is a basis with two vectors, so W has dimension 2.