Chapter 4 - Vector Spaces - 4.8 Row Space and Column Space - Problems - Page 325: 17

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Let $A$ be an $m × n$ matrix with $colspace(A) = nullspace(A)$ Then we have $\begin{bmatrix} a_{11}&. & . & . & a_{1n}\\ . & . & . & . & .\\ .& . & . & . & .\\ a_{m1} & . & . & . & a_{mm} \end{bmatrix}$ We can see that $Rowspace(A)$ is a subspace of $R^n$ and $colspace(A)$ is a subspace of $R^n$ Thus, $rowspace(A)=colspace(A)$ $Rowspace(A)$ is also subspace of $R^m$ and $colspace(A)$ is a subspace of $R^n$, if and only if $m=n$