Vector space of the product of two matrices

[redjoker]

I'm trying to prove (as part of a larger proof) that the product of a m x n matrix M with column space R^m and a n x o matrix N with column space R^n, MN, has column space R^m. I'm not sure where to begin. What I'm thinking should be the right approach is to show that any solution to M augmented with a vector v = (a_1, ..., a_m) can be tweaked to be a solution for MN, though I haven't been able to get there. Any suggestions?

 

[Defennder]

Did you really mean column space R^m for mxn matrix M and col space R^n for nxo matrix N ? Or was it supposed to be R^n for the former since matrix M has n columns?

 

[redjoker]

Yeah I meant column space. The assumption is that m,n,o form a non-decreasing sequence.