# Math Help - linear algebra proofs

1. ## [PROVED] linear algebra proofs

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

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

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

2. it's urgent. Can anyone help???

3. I proved both of them.
Thank you anyway.