I have a Field K and a k-vector space Homorphismus, $ \phi : {V} \times {W}$ and ${n \in N_0}$ and a n-Tupel $({s_1} ,\dots, {s_n})$ in ${V}$ (a) $(\phi(s_1,\dots ,\phi(s_n))$ $\textbf{linearly independent}$ in $ {V}$ $\Rightarrow$ $({s_1}, \dots ,{s_n})$ $\textbf{linearly independent}$ in ${V}$
(b) I have a Filed K and a Set X ,${n,m \in N_0}$,and a n-Tupel $({f_1} ,\dots, {f_n})$ in ${Map(X,K)}$, and a m-Tupel $({x_1} ,\dots, {x_m})$ in ${X}$ then :
$((f_1(x_1),\dots,f_1(x_m)),\dots,(f_n(x_1),\dots,f_n(x_m)))$ $\textbf{linearly independent}$ $\Rightarrow$ $({f_1} ,\dots, {f_n})$ $\textbf{linearly independent}$ in ${Map(X,K)}$,
to answer (b) I did define a vector space Homorphismus structure and used (a) let :
$\phi: Map(X,K)\rightarrow {K^m} , f_i \mapsto (f_i(x_1),\dots,f_i(x_m)) $ and I did prove that my structure can be vectorspace Homorphismus and then used to (a) to prove (b) ( is that correct ? )
(c) let $ k \in Z$, $f_k: R \rightarrow R ,x\mapsto \sin(kx)$ and $g_k: R \rightarrow R ,x\mapsto \cos(kx)$
the Question is if $(f_1,f_2,g_1,g_2)$ $\textbf{linearly independent}$ in Map(R,R) ?
$\textbf{my idea}$
to prove (c) i will use (b) , let m=3 and $x_1=\pi$ and =$x_2= 2x_1$, and $x_3=3x_1$
then i must prove that $((\sin(\pi),\sin(2\pi),\sin(3\pi),(\sin(2\pi),\sin(4\pi),\sin(6\pi)(\cos(\pi),\cos(2\pi),\cos(3\pi)),(\cos(2\pi),\cos(4\pi),\cos(6\pi)))$ =((0,0,0),(0,0,0,0),(-1,1,-1),(1,1,1)) and it is $\textbf{linearly independent}$ then according to (b) is $(f_1,f_2,g_1,g_2)$ $\textbf{linearly independent}$ in Map(R,R).
Is my idea correct ? in (b) and (c)