Row Reduction - Spanning Vectors

Question

How can row reduction be used to determine whether n vectors in R^m span R^m? Write down a precise answer using a theorem.

So far I know that no fewer than m vectors must span R^m, so this means that the number of specified vectors will only span the given space if n=m

Answer 1

The idea is

• to span $\mathbb{R}^m$ we need $m$ linearly independent vectors
• thus in the RREF we need to find $m$ pivot that is $rank=m$ and thus $dim (Im)=m$ by rank-nullity theorem