## linear algebra proofs

Question 1:
Let $v_{1},...v_{n} \in R^{n}$. If $S$ is a maximal linearly independent subset of $v_{1},...v_{n}$, show that each vector $v_{i}$ is a linear combination of the vectors in $S$.

It is clear if $v_{i} \in S$. How about $v_{i} \notin S$?

Question 2:
Show that $rank(A+B) \leq rank(A)+rank(B)$ for every $A,B \in M_{m,n}(R)$.

We know that $rank(A+B) \leq m$ and $rank(A)+rank(B) \leq 2m$. Can I conclude from here?

Question 3:
Let $u_{1},...u_{k},v_{1},...v_{n} \in R^{n}$. If each vector $u_{i}$ is a linear combination of vectors $v_{1},...v_{n}$. Show that the maximum number of linearly independent vectors in $u_{1},...,u_{k}$ cannot exceed the maximum number of linearly independent vectors in $v_{1},...v_{n}$.

I dont know where to start here.