TITLE: Is what I have done to Show that each element of $(E_h)$ is written as a linear combination of the functions $(\phi)_ i$?
QUESTION [0 upvotes]: Let $\phi_{i}$ are continious over mesh every integer, and define $E_h=\text{vect}\{\phi_i, i=1,\cdots N\}$
I want to Show that each element of $(E_h)$ is written as a linear combination of the functions $(\phi)_ i$, and that these functions are linearly independent ?
Attempt: It is known that finit elements based on discret space of globally  continious function and affine over each mesh , I have defined this functional space :
$V_h=\{ v\in C([0,1]) \text{such that}:v|_{[x_j,x_{j+1}]} ,0\leq j\leq n \quad \text{and}\quad v(0)=v(1)=0\}$, Now we have for every $v\in V_h$ is defined with unique way as:$v_h(x)=\sum_{j=1}^{n}v_h(x_j)\phi_j(x),\forall x\in [0,1]$, Now we have immediately :$\phi_j(x_i)=\delta_{ij}$ which it is the Kronecker symbole it is $1$ if $i=j$ and $0$ else , but am not sure if I have showed each element of $(E_h)$ can be  written as a linear combination of the functions $(\phi)_ i$, and that these functions are linearly independent

REPLY [1 votes]: If you consider the possibility that $a_1\phi_1 + \ldots a_n\phi_n =0$, then add the fact that $\phi_i(x_j) =\delta_{ij} $, we find that all the $a_i$'s must be zero, and therefore the set is linearly independent.
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