For each natural number n, let X be the set of polynomials of degree equal to n. Prove or give a counterexample: the set X is a vector space.

My teacher says it is NOT a vector space. I just need a little help proving it.

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    $\begingroup$ If it is degree equal to $n$ (and not smaller or equal), then have a look at the axioms that need to be satisfied for something to be a vector space. Specifically, is there an identity element of addition (see en.wikipedia.org/wiki/Vector_space)? $\endgroup$ – Shinja Apr 24 '17 at 1:04

Let $p(x)$ be a polynomial belonging to X

In order for X to be a vector space, $-p(x)$ belongs to X

But vector spaces are closed under addition, and this would imply that

$p(x)+(-p(x))=0\in X$

But this is a polynomial of degree zero, which is a contradiction.


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