I got an assignment which I just can't find the right way to solve. 

It goes like this: 

Let $\Omega \in R^n$ be a domain and $b\_1,...,b\_n:\Omega \to R$ smooth mappings (or functions, don't know the correct translation into english), so that for every $x\in\Omega$ the vectors $b\_1(x),...,b\_n(x)$ linear independent. 

Let $c\_1, ... , c\_n:\Omega\to R$ be mappings (or functions). 

Show that the function $F(x):=c\_1(x)b\_1(x)+...+c\_n(x)b\_n(x)$ is smooth when $c\_1,...,c\_n$ are smooth.