If we have a square matrix thats invertible, do its row and column space coincide? If we have a square matrix thats invertible, do its row and column space coincide?
Regarding an nxn invertible matrix:
-The row space of the matrix is R^n
-The column space of the matrix is R^n
-The rank of the matrix is n
Is this a sufficient way of proving the question, or am I missing something?

$\newcommand{\Reals}{\mathbf{R}}$The row space and column space of an $n \times n$ matrix are not generally equal, e.g.,
$$
A = \begin{pmatrix}
0 & 1 \\
0 & 0
\end{pmatrix},\quad
\text{row space} = \{0\} \times \Reals,\quad
\text{column space} = \Reals \times \{0\}.
$$
The row space and column space of an $n \times n$ matrix do always have the same dimension, however, and if this dimension is $n$, then each space is equal to $\Reals^{n}$.
