Thread: Linear Independence and Basis

1. 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.

2. For (2) you could take the Wronskian and show that it is not equaled to $0$.

$\det \begin{bmatrix} e^{rt} & e^{st} \\ re^{rt} & se^{st} \end{bmatrix} = se^{rt+st}- re^{rt+st} \neq 0$ (assuming that $s \neq r$) which implies that they are linearly independent.