Question

Let
V=\left(\begin{matrix}cos t \\ sin t\end{matrix}\right)

and define
\left(\begin{matrix}cos t_1 \\ sin t_1\end{matrix}\right) ⨁ \left(\begin{matrix}cos t_2 \\ sin t_2\end{matrix}\right)=\left(\begin{matrix}cos(t_1 + t_2) \\ sin (t_1+t_2)\end{matrix}\right)

c ⨀ \left(\begin{matrix}cos t \\ sin t\end{matrix}\right)= \left(\begin{matrix}cos ct \\ sin ct\end{matrix}\right)


Show that if ⊕ and ⨀ are the standard componentwise operations, then V is not a vector space.