If the columns of an $n \times n$ matrix are linearly independent, then the columns span $\mathbb{R}^{n}$ My textbook says that its true but I can't find a proof of this on the internet.

"If $A$ is an $n \times n$ matrix with linearly independent columns, then the columns of $A$ span $\mathbb{R}^{n}$."


The idea is that, if this matrix is called $A$, and the columns are linearly independent, then $A$ is invertible. This means given any system $$Ax=b$$ we can solve for $x$ simply by multiplying by $A^{-1}$, so $$x=A^{-1}b$$
Since $b$ could have been any vector in $\mathbb{R}^n$, the columns span $\mathbb{R}^n$, because $Ax$ is just a linear combination of the columns of $A$.
