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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?

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    After that, prove $T(a+b)=T(a)+T(b)$ and $T(ra)=rT(a), r\in \mathbb{R}$. That's assuming $V$ is a vector space of real polynomials-you didn't specify.2012-10-17

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