Differentiation of Exponential of a x by Cosine of b x and Exponential of a x by Sine of b x wrt x as Invertible Matrix

Theorem
Let $a, b, x \in \R$ be real numbers.

Suppose $a \ne 0 \ne b$.

Denote $\ds f_1 = \map \exp {a x} \map \cos {b x}$, $f_2 = \map \exp {a x} \map \sin {b x}$.

Let $\map \CC \R$ be the space of continuous real-valued functions.

Let $\struct {\map {\CC^1} \R, +, \, \cdot \,}_\R$ be the vector space of continuously differentiable real-valued functions.

Let $S = \span \set {f_1, f_2} \subset \map {\CC^1} \R$ be a vector space.

Let $D : S \to S$ be the derivative $x$.

Then, with respect to basis $\tuple {f_1, f_2}$, $D$ is expressible as:


 * $\mathbf D = \begin{bmatrix}

a & -b \\ b & a \end{bmatrix}$

and is invertible.

Proof
Thus:

Furthermore:


 * $\map \det {\mathbf D} = a^2 + b^2$.

By Matrix is Invertible iff Determinant has Multiplicative Inverse, $\mathbf D$ is invertible.