Question 1:

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

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

Question 2:

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

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

Question 3:

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 $\displaystyle v_{1},...v_{n}$. 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 $\displaystyle v_{1},...v_{n}$.

I dont know where to start here.