Linear Independence problem.

Hi, I have this question to solve.

Let f=exp(t), g=t, and h= 2+3t. Are f, g, and h linearly independent or linearly dependent?

I'm confused as to how I should figure this out. I understand that a set of vectors to be linearly independent, say v1=(1,2) and v2=(2,3), that the only solution for the system a+2b=0, 2a+3b=0 is for a=b=0 (i don't know if that is the case in the example i'm given). However, i'm confused as to how to go about the given problem. Thank you.

Re: Linear Independence problem.

It's really the same idea. Suppose a, b and c are such that af + bg + ch =0. This is really a function equation. That is it means __for all__ t, af(t) +bg(t) + ch(t) = 0. Now one way to prove independence is to find 3 different t values to get 3 equations in a, b and c whose only solution is 0. If you happen to know about Wronskians, there's another approach.

Re: Linear Independence problem.

Thank you for the reply.

So, I set the equation a*exp(t)+b*t+c(2+3t)=0 and I have to figure if the only solution to it is a=b=c=0 or not?

Re: Linear Independence problem.

Yes, the only solution for which the equation holds __for all__ t is a=b=c=0. So if you can find 3 explicit t values which then force a=b=c=0, then the function equation is 0 for all t only if a=b=c=0.

Re: Linear Independence problem.

Re: Linear Independence problem.

As a "function" equation, that has to be true for all t. So setting t equal to three different numbers will give you three different equations to solve for a, b, and c. For example, it t= 0, you get a(exp(0))+ b(0)+ c(2+ 3(0)= a+ 2c= 0.

Now, set t equal to, say, 1 and -1. What equations do you get?