Let $V_i = 1,...,N$ be a collection of vector spaces over a field $F$. Consider the Cartesian product $V=V_1 \times V_2 \times ... \times V_N$ with the natural projections $\pi=V \rightarrow V_i$. Prove that $\pi$ is linear and compute its kernel.
I am pretty confused as to what I need to do for this question. So far I've chosen an arbitrary basis for $V$ but I don't know what to do after that.