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

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

\[c \odot (x,y) = \left(c x,\,c y - 3 \, c + 3\right) .\]

(a) Show that scalar multiplication is associative, that is:

\[a\odot(b\odot (x,y))=(ab)\odot(x,y). \]

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