# Thread: linearly independency in function space

1. ## linearly independency in function space

Let V be the vector space of all real-valued continuous functions. Is the following set linearly dependent? If yes express one vector as a linear combination of the rest.

$\displaystyle {cos(t), sin(t), e^t}$

I'm not sure how to get the equations to form a matrix, I don't see any homogeneous system here.

2. Try evaluating the Wronskian. What does that give you?

3. Is your set of functions linearly dependent ?

4. We can write $\displaystyle a_1 \cos t+a_2\sin t+a_3e^t =0$ and try to see if we must have $\displaystyle a_1=a_2=a_3=0$. What does this equation give if $\displaystyle t=0$? If $\displaystyle t=\pi$? If $\displaystyle t=2\pi$?

5. $\displaystyle \left ( \cos(t),\sin(t) \right )$ is obviously l.independent.
to prove that $\displaystyle \left ( \cos(t),\sin(t),\exp(t) \right )$ is dependent we must prove that
$\displaystyle \exp(t) \in \texttt{Span}(\cos(t),\sin(t))$.
Now suppose we have $\displaystyle \exp(t) \in \texttt{Span}(\cos(t),\sin(t))$
$\displaystyle \exp(t) \in \texttt{Span}(\cos(t),\sin(t))\Rightarrow \exp(t)=a\cos(t)+b\sin(t)$
for $\displaystyle t=\frac{\pi}{2}$ we have $\displaystyle b=\exp(\frac{\pi}{2})$
for $\displaystyle t=0$ we have $\displaystyle a=1$
hence $\displaystyle \exp(t)=\cos(t)+\exp(\frac{\pi}{2})\sin(t)$ which is not true.
so $\displaystyle \left ( \cos(t),\sin(t),\exp(t) \right )$ must be l.independent.