# Linear Independence and Basis

• Oct 10th 2007, 11:56 PM
mathgirl
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):mad:
b. Prove that S contains at least n vectors.
• Oct 11th 2007, 12:16 AM
tukeywilliams
For (2) you could take the Wronskian and show that it is not equaled to $\displaystyle 0$.

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