If $y_1(x) , y_2(x) ,\ldots,y_n(x)$ are linearly independent in $C[b,c]$ then they are Linearly Independent in $C[a,d]$, where $a<b<c<d.$
So I know if the Wronskian isn't zero for at least one point then the functions are Linearly Independent, the converse isn't true , is it true. If the wronskian isn't zero at the subinterval for some $x$ , then function would be Linearly Independent at the interval .
Can we consider that $C[a,d]$ is a subspace of $C[b,c]$ ? So if the functions are Linearly Independent in the vector-space , they would be Linearly Independent in the subspace?
I'm confused, I feel that the statement is false, but I haven't found any counterexamples yet.
Linear dependence/independence is a feature that must be true at every point of your space. When you say that $y_i$ are linearly independent on a set $Z$, that means there doesn't exist a fixed set of $a_i$ (at least one which is nonzero), such that:
$$a_1y_1(x)+\cdots+a_ny_n(x)=0, \forall x\in Z.$$
So suppose that $y_i$ are linearly independent on $[b,c]$ but not on $[a,d]$. This means there exist $a_i$ such that:
$$a_1y_1(x)+\cdots+a_ny_n(x)=0, \forall x\in [a,d].$$
This would imply that the same $a_i$ satisfy:
$$a_1y_1(x)+\cdots+a_ny_n(x)=0, \forall x\in [b,c],$$
which in turn implies linear dependence on $[b,c]$, a contradiction.