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Exercise 4

Avg difficulty 3.0

In Pn\mathcal{P}_{n}, the even polynomials are the members of this set

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

and the odd polynomials are the members of this set.

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

Show that these are complementary subspaces.

Hefferon, Linear Algebra, Vector Spaces, Combining Subspaces, Exercise 4

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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.