Linear Independence

When comparing two functions, say $f\_1$ and $f\_2$, in many applications it is important to know if they are in no way constant multiples of each other, so linearly independent such that the only constants $c\_1$ and $c\_2$ satisfying the equation $c\_1f\_1(x)+...+c\_nf\_n(x)=0$ are $c\_1=...=0$.

Wronskian Test

If the Wronskian of this set of functions is $W\ne 0$ at least at one point, the functions are linearly independent.

$$W(f\_1,f\_2)(x)=\left|\begin{array}{cc}f\_1(x)& f\_2(x)\\ f^{\prime }\_1(x)& f^{\prime }\_2(x)\end{array}\right|$$

• If $det\_W\ne 0$ at least one point in the interval $f\_1,f\_2$ are linearly independent.
• $det\_W=0$ the test is inconclusive