Show that the set of all functions E CR) such that df/dt exists anddf / dt = 0 is a one-dimensional subspace of C(R).ThenGeneralize the result. For example, what is the dimension of thesubspace consisting of all f such that df /d? = 0?

Let W=f∈CR/f'=0. Let f,g∈W and α∈R. Then αf+g=αf+gx=αfx+gx=αfx+gx⇒αf+g'=αf'+g'=0+0, since f,g∈W=0…

Answered: Show that the set of all functions./ E…

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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Show that the set of all functions E CR) such that df/dt exists anddf / dt = 0 is a one-dimensional subspace of C(R).
ThenGeneralize the result. For example, what is the dimension of thesubspace consisting of all f such that df /d? = 0?

Expert Solution

Step 1

Let $W = \{f \in C (R) / f' = 0\}$ .

Let $f, g \in W$ and $α \in R$ .

Then

$\begin{array}{rcl} α f + g & = & (α f + g) (x) \\ = & (α f) x + g (x) \\ = & α f (x) + g (x) \\ \Rightarrow & (α f + g)' = α f' + g' \\ = & 0 + 0, \sin c e f, g \in W \\ = & 0 \end{array}$

Hence $α f + g \in W$ . Therefore W is a subspace of C(R).

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