I want to show that the family of hat functions $\Phi _i$ is a basis of the following space \begin{equation} V_h={v _h\in C(\overline{\Omega})\ such\ that\ v_h(0)=v_h(1)=0 \forall j \in\ (1,...,N)\ and\ v_h\ is\ linear\ on\ the\ closed\ interval\ from\ x_j-1\ to \ x_j} \end{equation}
I know that I have to show linear independence and that each element of $V_h$ can be written as a linear combination of hat functions. I tried to show the linear independence by assuming that $\sum\Phi _i(x)c_i=0$. Then I tried to show that $c_i=0 \forall i$. I think using the fact that $\Phi _i$ is only nonzero on the interval $[x _{i-1},x_i]$ might be helpful to show that $c_i=0$ for all i. However, I do not know how to use it.
After that I wanted to prove that each element can be expressed as a linear combination i.e. that every $v_h$ can be written as $v_h=\sum \Phi _i c_i$. Unfourtunately, I do not have any idea how I could prove this statement.
Could anybody explain it to me? Thanks!