Consider the family of functions $f_n$ and $(g_n)_{n\in\mathbb{N}}$, where for each $n$ in $\mathbb{N}$:
- $f_n: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f_n(x) = x^n \cos(x)$,
- $g_n: \mathbb{R} \rightarrow \mathbb{R}$ defined by $g_n(x) = x^n \sin(x)$.
Additionally, let there be a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the range of $f$ is infinite.
The task is to examine whether the combined family $f_n \cup (g_n)_{n\in\mathbb{N}}$ is independent in the vector space of real-valued functions.
My initial thoughts are to approach this problem by checking for linear independence among the functions. I believe that the functions $x^n \cos(x)$ and $x^n \sin(x)$ might be independent due to their differing structures, especially considering the powers of $x$ coupled with trigonometric functions. However, I am not entirely sure how to formally prove this independence, especially since the function $f$ with an infinite range introduces additional complexity to the problem.
My try :
To show independence, we consider a linear combination of these functions set to the zero function: \begin{equation} a f_n(x) + b g_n(x) = 0 \quad \forall x \in \mathbb{R}, \end{equation} where $a$ and $b$ are constants, and we wish to show that this implies $a = b = 0$.
As I am just starting out with algebra, I find myself struggling to transform these intuitions into a rigorous mathematical proof. Could someone please guide me on how to approach this problem or suggest resources that could help me understand how to prove the independence of such function families?