TITLE: Showing $1,e^{x}$ and $\sin{x}$ are linearly independent in $\mathcal{C}[0,1]$
QUESTION [10 upvotes]: 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.

REPLY [2 votes]: You have $a + b \sum_{k\geq 0} \frac{x^k}{k!} + c \sum_{k\geq 0} (-1)^{k} \frac{x^{2k+1}}{(2k+1)!} = 0$
All the coefficients must be 0 above, so, for the constant term we have $a+b = 0$ and comparing coefficients of $x$ we have $b+c=0$, comparing coefficient of $x^2$ we have $b=0$, from which it follows $a=b=c=0.$