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.