Question 1:

Let . If is a maximal linearly independent subset of , show that each vector is a linear combination of the vectors in .

It is clear if . How about ?

Question 2:

Show that for every .

We know that and . Can I conclude from here?

Question 3:

Let . If each vector is a linear combination of vectors . Show that the maximum number of linearly independent vectors in cannot exceed the maximum number of linearly independent vectors in .

I dont know where to start here.