We have seen several examples of bases in different vector spaces. In this subsection, and the next (Subsection B.BNM), we will consider building bases for $\complex{m}$ and its subspaces.
Suppose we have a subspace of $\complex{m}$ that is expressed as the span of a set of vectors, $S$, and $S$ is not necessarily linearly independent, or perhaps not very attractive. Theorem REMRS says that row-equivalent matrices have identical row spaces, while Theorem BRS says the nonzero rows of a matrix in reduced row-echelon form are a basis for the row space. These theorems together give us a great computational tool for quickly finding a basis for a subspace that is expressed originally as a span.
Example IAS provides another example of this flavor, though now we can notice that $X$ is a subspace, and that the resulting set of three vectors is a basis. This is such a powerful technique that we should do one more example.