Rank one Matrices

Every matrix $M$ from the set of all $m\times n$ matrices with rank $r$ can be broken down into a product of $r$ $m\times n$ matrices with rank one. They are the building blocks of every matrix $M$. For rank one matrices (only one independent column) the special definition arises, showing that the matrix $A$ is jsut a linear combination of $v^T=\text{[}a,b,\dots ]$ columns taken from the single column $u$ that makes up the entire basis of $A$. $v^T$ is therefore the basis for the row-space $C(A^T)$

$$A=u{v}^T$$