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Exercise 24

Avg difficulty 3.0

In Pn(R)\mathcal{P}_{n}(\mathbb{R}), the vector space (over the reals) of degree nn polynomials,

E={p(x)∈Pn(R)∣p(−x)=p(x) for all x}\mathcal{E}=\text{$\{\hskip 1.00006ptp(x)\in\mathcal{P}_{n}(\mathbb{R})\mid p(-x)=p(x)\text{\ for all\ }x\hskip 1.00006pt\}$}

and

O={p(x)∈Pn(R)∣p(−x)=−p(x) for all x}\mathcal{O}=\text{$\{\hskip 1.00006ptp(x)\in\mathcal{P}_{n}(\mathbb{R})\mid p(-x)=-p(x)\text{\ for all\ }x\hskip 1.00006pt\}$}

are the even and the odd polynomials; p(x)=x2p(x)=x^{2} is even while p(x)=x3p(x)=x^{3} is odd. Show that they are subspaces. Are they complementary? Are they invariant under the differentiation transformation?

Hefferon, Linear Algebra, Similarity, Jordan Canonical Form, Exercise 24

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Adapted from Linear Algebra © 2016 Jim Hefferon. Licensed under CC BY-SA 3.0 US. Source: Linear Algebra. Imported at revision df2262e.