Thread: Linear Independence and Basis

Thank you so much for taking time to read this. I'm in deep trouble so any help will be very appreciated.

(1) Prove that a set S of vectors is linearly independent iff each finite subset of S is linearly independent.

(2) let f,g in F(R,R) be the functions defined by f(t)= e^(rt) and g(t)= e^(st) where r does not equal s. Prove that f and g are linearly independent in F(R,R).

(3) Let V be a vector space having dimension n, and let S be a subset of V that generates V.
a. Prove that there is a subset of S that is a basis for V. (You cannot assume that S is finite)
b. Prove that S contains at least n vectors.

For (2) you could take the Wronskian and show that it is not equaled to .

(assuming that ) which implies that they are linearly independent.