Let $f(x) = x^{4} - 1$ and $g(x)= x^{4}- 4$.
Let $V$ be the vector space consisting of real polynomials of degree less than or equal to $3$.
Let $T:V \longrightarrow V$ be the map such that for $h \in V$, $T(h)$ is the remainder of the Euclidean division of $fh$ by $g$.
Show that $T$ is a linear map.
I thought I could start by proving that $T(0) = 0$? How would I proceed after that?