[Math] If $n$ vectors are linearly independent, is their span $\mathbb{R}^n$

Question

Have $n$ vectors in $\mathbb{R}^n$.

If the $n$ vectors are linearly independent, can we conclude that their span is $\mathbb{R}^n$?

Answer 1

You can use the theorem that every linearly independent set in a vector space $V$ can be extended to a basis of $V$. Since the dimension of $\mathbb{R}^n$ is simply $n$, the extension of the $n$ vectors to a basis is trivial (i.e. the vectors are unchanged); these vectors therefore already span $\mathbb{R}^n$.