# Prove that if $S$ is a change of basis matrix, its columns are a basis for $\mathbb{R}^n$

Let's say we have a basis $B$ of $\mathbb{R}^n$ consisting of vectors $\vec{v}_1$ through $\vec{v}_n$, and some other basis $C$ of $\mathbb{R}^n$. Then, would $[\vec{v}_1]_C$ through $[\vec{v}_n]_C$ be a basis for $\mathbb{R}^n$ as well? It seems obvious, but I am not sure how to go about this.

-