# Thread: Matrix inequality ( column spaces )

1. ## Matrix inequality ( column spaces )

Matrix A is an mxn matrix with rank r and matrix B is a MxN matrix with rank R. The column space of A is in or equal to the column space of B.

Fill in the inequality ( <= , < , >, = , >= )

1) m ____ M
2) r _____ R

I know that for number two it must be <= but I'm not sure about #1.

I'm thinking it's = but then I'm doubting it and thinking it might be <=

2. ## Re: Matrix inequality ( column spaces )

I assume you mean: Let $V_A$ be the column space of A and $V_B$ the column space of B with $V_A\subseteq V_B$.

Thus every column vector of A must be a column vector of B. So it must be that m=M.