SOLUTION: Let f,g, element F(R,R) be the function defined by f(t)=e^rt and g(t)=e^st, where r cannot equal s. prove that f and g are linearly independent in F(R,R)

Algebra ->  Algebra  -> College  -> Linear Algebra -> SOLUTION: Let f,g, element F(R,R) be the function defined by f(t)=e^rt and g(t)=e^st, where r cannot equal s. prove that f and g are linearly independent in F(R,R)      Log On


   

Question 346626: Let f,g, element F(R,R) be the function defined by f(t)=e^rt and g(t)=e^st, where r cannot equal s. prove that f and g are linearly independent in F(R,R)
Answer by Fombitz(13828) About Me  (Show Source):
You can put this solution on YOUR website!

By the zero product property,
e%5E%28rt%29=0 or 1++%2B-+e%5E%28%28s-r%29t%29=0
The first case can never happen since
e%5E%28rt%29%3E0 for all values of rt
The second case can only happen when
e%5E%28%28s-r%29t%29=1
When r=s.
Since this is not the case, then f and g are linear independent.
f%28t%29%2Bg%28t%29=0 only when f%28t%29=g%28t%29=0