Let A be an mxn matrix of rank 1. Using the definition of rank, show that there exists an mx1 matrix X and an 1xn matrix Y such that A=XY

Question

How do I approach this question? I know the X and Y can be the basis for the columnspace of A and the basis of rowspace of A, but how can it be? Thanks for help!

Answer 1

If the rank of $A$ is equal to $1$, then its columns are all in the vectorial line generated by a non zero vector $Y$ of dimension $m$.

That means that the columns of $A$ are $$x_1 Y, x_2 Y, \dots , x_nY.$$ If you denote $X^T=(x_1, \dots , x_n)$, you get the desired result

$$A=X^T Y.$$