I want to show that $\displaystyle rank(A+B) \leq rank(A)+rank(B)$ where $\displaystyle A,B \in M_{m,n}(R)$.

It is obvious that $\displaystyle rank(A), rank(B), rank(A+B)$ are $\displaystyle \leq m$. So, $\displaystyle rank(A)+rank(B) \leq 2m, rank(A+B) \leq m$. Can I conclude now that $\displaystyle rank(A+B) \leq rank(A)+rank(B)$? I think there should be somemore steps to continue this proof.

Next questions, I need some hints.

Let $\displaystyle u_{1},...,u_{k},v_{1},...v_{n} \in R^{n}$. If each vector $\displaystyle u_{i}$ is a linear combination of vectors of $\displaystyle v_{1},...,v_{n}$. Show that the maximum number of linearly independent vectors in $\displaystyle \{u_{1},...,u_{k}\}$ cannot exceed the maximum number of linearly independent vectors in $\displaystyle \{v_{1},...,v_{n}\}$