Exponential of a x by Cosine of b x and Exponential of a x by Sine of b x are Linearly Independent

Theorem
Let $x, a, b \in \R$.

Then $e^{a x} \map \cos {b x}$ and $e^{b x} \map \sin {b x}$ are linearly independent.

Proof
Let $\alpha_1, \alpha_2 \in \R$ be such that:


 * $\alpha_1 e^{a x} \map \cos {b x} + \alpha_2 e^{a x} \map \sin {b x} = 0$

Suppose $x = 0$.

Then $\alpha_1 = 0$.

Suppose $\ds x = \frac \pi {2 b}$.

Then:


 * $\ds \alpha_2 \map \exp {\frac {\pi a} {2 b}} = 0$

Hence, $\alpha_2 = 0$.

The conclusion follows from the definition of linearly independent real functions.