= Column space
{title2=$\operatorname{col}A$}
The column space of a <matrix> $A$ is the <linear span> of its columns. It is the <image of a linear map> represented by $A$.
If an $m\times n$ <matrix> has columns $c_1,\ldots,c_n$, then <matrix multiplication> gives $Ax=x_1c_1+\cdots+x_nc_n$. Thus its <column space> consists of exactly the attainable output <vectors> in $\mathbb F^m$, and $Ax=b$ has a solution exactly when $b$ belongs to this <column space>. Its <dimension> is the <matrix rank>; the <kernel> instead describes input <vectors> in $\mathbb F^n$ whose contributions cancel.<Interactive Linear Algebra by Dan Margalit and others/Solution Sets>[Interactive Linear Algebra, §2.4.3, Solution Sets and Column Spans: attainable outputs and input solution sets occupy different spaces]{ref}
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