Checking if $m$ vectors span an $\mathbb{R}^n$ dimensional vector space using reduced row echelon form

Question

Let $S=\{v_i\}$, $i \in $ {1,2...m}. We want to see if $S$ spans $\mathbb{R}^n$ where $m>n$

I am guessing that in order to check if $S$ spans $\mathbb{R}^n$, we can put all the vectors in $S$ as columns vectors in a matrix

$[v_1 v_2 ... v_m]$

Then find the reduced row echelon form of that matrix. If there are any row vectors that are equal to $\vec{0}$, then $S$ does not span $\mathbb{R}^n$. Otherwise, it does.

I tested it with some matrices and it seems to work but I am not sure if works in all cases.

Question: Does this algorithm work in all cases? If not, what cases would it not work in?

Answer 1

If there are zero rows, it means the rank is less than $n$, and hence it can't span $\mathbb{R}^n$.

If there are no zero rows, it means the rank is $n$. Hence it spans $\mathbb{R}^n$.