- Column space / Image: $C(A)$, since it corresponds to $C(A) \equiv \{ y : \exists x, y = Ax \}$
- Null space $N(A) \equiv \{ k : Ak = 0 \}$.
- Row space: row spans the row space, so it's all linear combinations of the rows of $A$. This is the same as all combinations of the columns of $A^T$. Row space is denoted by $C(A^T)$.
- Null space of $A^T$: $N(A^T)$, also called as the "left null-space of $A$".

- The dimension of the column space is the rank $r$.
- The dimension of the row space is also the rank $r$.
- The dimension of the nullspace is $n - r$.
- Similarly, the left nullspace must be $m - r$.

$\begin{aligned}
&M [AI] = [RE] \\
&MA = R; MI = E \implies M = E
\end{aligned}$

So the matrix that takes $A$ to $R$ is $E$! We can find the basis for the left
nullspace by lookinag at $E$, because $E$ gives us $EA = R$.