I want to show linear independence in the wronskian implies linear independence between the functions $f_1(x)$, $f_2(x)$, $f_3(x)$. Let $f_1(x)$, $f_2(x)$, $f_3(x)$ be real-valued functions with first and second order derivatives on the interval $(a, b)$. Consider the following:
$$W(x)=\left( \begin{array}{ccc} f_1(x) & f_1'(x) & f_1''(x) \\ f_2(x) & f_2'(x) & f_2''(x) \\ f_3(x) & f_3'(x) & f_3''(x) \end{array} \right)$$
Formally, if $\exists x$ $\in$ $(a, b)$ s.t. the rows of $W(x)$ are linearly independent then $f_1(x)$, $f_2(x)$, $f_3(x)$ are linearly independent.
Suppose $c_1f_1+c_2f_2+c_3f_3=0$ has only the zero solution. Differentiate twice and write the three equations as a single matrix equation. Since the matrix admits only the zero solution the determinant of the coefficient matrix must be nonzero. This coefficient matrix is precisely the Wronskian's matrix. ( I usually call the determinant of the matrix you write the Wronskian )