if $A$ is an $ n × n$ matrix, the A is the change of coordinates matrix from some bases $B$ and $C$ of $\mathbb R^n$

Question

Question:

Prove the following: if $A$ is an invertible $n × n$ matrix, then A is the change of coordinates matrix from some basis $B$ of $\mathbb R^n$ to some basis $C$ of $\mathbb R^n$

I know this is true, but I have no idea how to even start solving this.

Answer 1

Fix your favorite basis of $\mathbb R^n$, such as the standard basis consisting of the columns of $\boldsymbol I_n$. Consider what $\boldsymbol A$ does to this basis: $$\boldsymbol A \boldsymbol I_n = \boldsymbol A.$$ You can see that the standard basis elements are mapped to the columns of $\boldsymbol A$. These vectors are linearly independent because $\boldsymbol A$ is invertible.