[See Steps] Let \mathscrF denote the vector space of real-valued functions on the real line. Show that sin x, sin (2 x), and sin (3 x) are linearly independent

Question: Let $\mathscr{F}$ denote the vector space of real-valued functions on the real line.

Show that $\sin x, \sin (2 x)$, and $\sin (3 x)$ are linearly independent in $\mathscr{F}$.
Let $V$ be the span of $\sin x, \sin (2 x)$, and $\sin (3 x)$ in $\mathscr{F}$, so that these three functions form a basis of $\mathrm{V}$. Show that $(\sin x)^{3}$ is in $V$ and find its coordinates with respect to the basis $(\sin x, \sin (2 x), \sin (3 x))$.

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[See Steps] Let \mathscrF denote the vector space of real-valued functions on the real line. Show that sin x, sin (2 x), and sin (3 x) are linearly independent

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