Please show work **Problem 13:**Let $ M $ be the set of all vectors (or polynomials) $ x $ in $ \mathbb{P} $ (over $ \mathbb{R} $) for which $ x(t) = x(-t) $ — even polynomial functions — holds identically in $ t $. Show that $ M $ is a subspace of $ \mathbb{P} $ (over $ \mathbb{R} $).

Answered: Let M be the set of all vectors (or…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Please show work

**Problem 13:**

Let $ M $ be the set of all vectors (or polynomials) $ x $ in $ \mathbb{P} $ (over $ \mathbb{R} $) for which $ x(t) = x(-t) $ — even polynomial functions — holds identically in $ t $. Show that $ M $ is a subspace of $ \mathbb{P} $ (over $ \mathbb{R} $).

Expert Solution

Step 1

Given $M$ be the set of all vectors $x$ in $P (R)$ for which $x (t) = x (- t)$ .

A non empty subset $M$ of a vector space $V$ , is said to be subspace of it, if it is a vector space over same field and same binary operations. Since elements of the subset $M$ are also elements of $V$ , therefore they satisfied all the axiom of vector spaces other then closeness property. So to check a non empty subset is subspace, it is enough to check that the subset is closed under vector addition and scalar multiplication.