> [!NOTE] Definition (Column span) > Let $A$ be a [[Real Matrices|real matrix]] with order $m\times n.$ Then column span of $A,$ denoted $\text{Colspan}(A),$ is the [[Span of Subset of Real n-Space|span]] of the [[Matrix Column|columns]] of $A$. # Properties By [[Image of Left Multiplication Linear Map of Real Matrix Equals Column Span of Real Matrix]], ... > [!NOTE] Lemma (Basis of Column span are the nonzero columns of RCEF) > Let $A\in\mathbb{F}^{mn}$. Then the nonzero columns of the [[Reduced Row Echelon Form for Real Matrix#^ffb67a|RCEF]] of $A$ form a [[Basis of Vector Space|basis]] of the column span of A. >*Proof*. Let $\underline{d}_{1},\dots,\underline{d}_{l}$ be the nonzero columns of the reduced column echelon form of $A$, where $l \leq n$. By reversing the column operations we recover the columns of $A$ as linear combinations of $\underline{d}_{1},\dots,\underline{d}_{l}$. Thus $\underline{d}_{1},\dots,\underline{d}_{l}$ span $\text{Colspan}(A)$. >Finally $\underline{d}_{1},\dots,\underline{d}_{l}$ are linearly independent since each of them has a pivot entry $1$ is some row where all the others have a zero entry. > [!NOTE] Definition (Column Rank) > The column rank or rank of $A$ is defined as the dimension of its column span (i.e cardinality of the basis of its column span). ^d6a7be > [!NOTE] Lemma (Column Rank is the same as number of nonzero columns of SNF) > The rank of [[Smith Normal Form for Real Matrix|Smith normal form]] is equal to the number of its nonzero columns and this is this equals the dimension of its column span. >See [[Reduced Row Echelon Form for Real Matrix#^f88761|proof]].