Check if two functions are linearly independent.

Question

I have to check if $f(s)=s,~g(s)=e^{ks},~k\in\mathbb{C}$ are linearly independent over $\mathbb{R}$.

The wronskian is $W[s,e^{ks}]=e^{ks}(ks-1)$. Then, if we take $s=1,~k=1+0i$, it becomes $0$ and hence, it is not linearly independent.

Is my reasoning correct?

Edit: thank you all. My error was that the Wronskian is used when those functions are solutions for a differential equation, and I don't have that hypothesis.

Answer 1

I don't know what Wronskian is but here is a way to show that $f, g$ is linearly independent.

Proof: Suppose that $f, g$ is linearly dependent. Without loss of generality, suppose that $g=a_1f$ for some non-zero $a_1\in R$. Then differentiating both sides twice we see that the left hand side is non-zero and the right hand side is zero. We get a contradiction and thus $f, g$ is linearly independent.

Edit: For the case of $k=0$, we can differentiate both sides once and get a contradiction as the right hand side would be equal to $a_1$ and $a_1\ne 0$.

Answer 2

$k$ is fixed and you don't get to choose it. Don't you have the result that if the Wronskian is nonzero for some value of $s$, then the functions must be linearly independent? After all, if they are linearly dependent, then the Wronskian will be zero for all values of $s$. In your case, the Wronskian will be zero for at most one value of $s$ (if $k$ happens to be nonzero and real).