Wronskian Theory

Question

Use the Wronskian theorem to prove the given functions are linearly independent on the indicated interval.

$e^x , e^{-x} , e^{4x}$; $(-\infty, \infty)$

Answer 1

A hint: Use $$D^k\bigl(e^{c x}\bigr)=c^k \ e^{cx}\qquad(c\in{\mathbb C}, \ k\in{\mathbb N}_{\geq0})$$ to set up the matrix needed for the Wronskian, and note that the determinant is a linear function of its column vectors.

Answer 2

You can also use the following good fact:

A set of functions $f_1(x),f_2(x),…,f_n(x), x\in I$ is linearly dependent on $I$ iff the determinant below is identically zero on $I$: $$ \det\left( \begin{array}{ccccc} \int_{a}^{b} f_1^2 dx& \int_{a}^{b} f_1f_2 dx&… &\int_{a}^{b}f_1f_ndx \\ \int_{a}^{b}f_2f_1dx & \int_{a}^{b}f_2^2 dx &... &\int_{a}^{b}f_2f_ndx \\ ⋮ & ⋮ & ⋮ &⋮ \\ \int_{a}^{b}f_nf_1dx & \int_{a}^{b}f_nf_2dx&... &\int_{a}^{b}f_n^2dx \end{array} \right) $$