Came across this problem during my studies... Here it is:

$\displaystyle f, g \in F(R, R)$ be the functions defined by $\displaystyle f(t)=e^{rt}$ and $\displaystyle g(t)=e^{st}$ where r is not equal to s. Prove that $\displaystyle f$ and $\displaystyle g$ are linearly independent in $\displaystyle F(R, R)$.

However, I'm confused about what I can allow $\displaystyle t$ to be. Because if $\displaystyle t=0$ then $\displaystyle e^{st}$ and $\displaystyle e^{rt}$ are both equal to 1.

Thus, I can form the equation $\displaystyle ae^{rt}+be^{st}=0$

If $\displaystyle t=0$ then letting $\displaystyle a=1$ and $\displaystyle b=-1$, the equation is true. Thus, aren't the equations linearly dependent?

EDIT: Forgot to include another problem I'm having trouble with!

Let u, v, and w be distinct vectors of a vector space V. Show that if {u, v, w} is a basis for V, then {u+v+w, v+w, w} is also a basis for V.

I have no idea how to even start that one!