Showing $1,e^{x}$ and $\sin{x}$ are linearly independent in $\mathcal{C}[0,1]$

How do i show that $f_{1}(x)=1$, $f_{2}(x)=e^{x}$ and $f_{3}(x)=\sin{x}$ are linearly independent, as elements of the vector space, of continuous functions $\mathcal{C}[0,1]$.

So for showing these elements are linearly independent, one needs to show that if $$ a_{1} \cdot 1 + a_{2} \cdot e^{x} + a_{3} \cdot \sin{x}=0$$ then from this we should conclude that $a_{1}=a_{2}=a_{3}=0$. But i am not being able to deduce this.

Solution 1:

If you differentiate your equation with respect to $x$ four times, you get that $a_2 e^x+a_3 \sin x = 0$, from which it follows that $a_1=0$. Now, letting $x=0$, you see that $a_2 e^x=0$, so $a_2=0$, and it follows readily that $a_3=0$.

With problems like this, always remember that you are dealing with functions. That means that you can treat them as functions as well: you can differentiate them and set their values equal to something.

Solution 2:

You may calculate the Wronskian of $f_1(x)$, $f_2(x)$, $f_3(x)$ and check that it doesn't vanish on $[0,1]$. This implies that the functions are linearly independent.

A caveat. If the Wronskian of a set of functions is equal to zero identically, it doesn't follow in general that the functions are linearly dependent.