The [[Column Space]] of some [[Matrix]] $A$ is the [[Span]] of the [[Set]] of column [[Vector|Vectors]] inside $A$. The [[Column Space]] is equal to the [[Set]] of all [[Image|Images]] produced by $As corresponding [[Linear Transformation]]. $\huge \let A \in M_{n\times m} $ $\huge \op{\mathrm{Im}}(T) = \op{col}(A) = \op{span}\pa{\vec A_{1}, \vec A_{2}, \dots, \vec A_{m}} $ >[!tip] >The column space of the [[Reduced Row Echelon Form|RREF]] of a [[Matrix]] is typically not the column space of the [[Matrix]]. >$\huge \op{col}(A) \neq \op{col}(\op{rref}(A)) $ $\huge \let Q: \mat{x_{1}\\x_{2}}\mapsto \mat{3x_{1}+x_{2}\\-x_{1}+x_{2}} $ $\huge C = \mat{3&1\\-1&1}$ $\huge \begin{align} \end{align} $