> [!NOTE] Lemma > Let $A$ be a [[Real Matrices|real matrix]] of order $m\times n.$ Let $AF$ be the [[Reduced Column Echelon Form for Real Matrix|RCEF]] of $A.$ Then their [[Column Span of Real Matrix|column spans]] satisfy $\text{Colspan }AF= \text{Colspan}A.$ **Proof**: follows from [[Elementary Column Operations Preserve Image of Left Multiplication Linear Map of Real Matrix]] that $\text{Im }L_{AF}=\text{Im }L_{A}$ and [[Image of Left Multiplication Linear Map of Real Matrix Equals Column Span of Real Matrix]] that $\text{Im }L_{A} =\text{Colspan }A.$