Let \(V\) be the set of all pairs \((x,y)\) of real numbers together with the following operations:

\[(x_1,y_1)\oplus (x_2,y_2)= \left(5 \, x_{1} + 5 \, x_{2},\,3 \, y_{1} + 3 \, y_{2}\right) \]

\[c \odot (x,y) = \left(c x,\,c y\right) .\]

(a) Show that scalar multiplication distributes over vector addition, that is:

\[c\odot \left((x_1,y_1)\oplus(x_2,y_2)\right)=c\odot(x_1,y_1)\oplus c\odot(x_2,y_2). \]

(b) Explain why \(V\) nonetheless is not a vector space.