Same rank implies same column space True or False? If $X$ and $Y$ are conformable matrices for the product $XY$ and if the rank of $XY$ equals the rank of $X$, then the span of the columns of $XY$ equals the span of the columns of $X$. If it is true, prove it. This seems like it can not be true, but I have not yet found a counterexample.

**True** : Note that a column of $XY$ is a linear combination of columns of $X$. Hence $$ {\rm col}\ (XY)\subset {\rm col}\ (X)$$ Since rank are same, they are same.