Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 89530 by mathocean1 last updated on 17/Apr/20

$$\mathrm{F}\mathrm{which}\mathrm{is}\mathrm{the}\mathrm{set}\mathrm{of}\mathrm{funtions}\mathrm{from}\mathbb{R}\mathrm{to}\mathbb{R}$$$$\mathrm{is}\mathrm{a}\mathrm{vectorial}\mathrm{space}\mathrm{and}\mathrm{G}\left(\mathrm{a}\mathrm{part}\mathrm{of}\mathrm{F}\right)\mathrm{is}$$$$\mathrm{the}\mathrm{set}\mathrm{of}\mathrm{odd}\mathrm{functions}\mathrm{such}\mathrm{as}$$$$\mathrm{G}=\{f\in \mathrm{F}/\mathrm{\forall }\mathrm{x}\in \mathbb{R},f\left(x\right)=-f(-x)\}$$$$1)S\mathrm{how}\mathrm{that}\mathrm{G}\mathrm{is}\mathrm{sub}\mathrm{vector}\mathrm{space}\mathrm{of}\mathrm{F}$$$$\mathrm{in}\mathbb{R}.$$

Commented by arcana last updated on 18/Apr/20

$$\mathrm{The}\mathrm{function}\mathrm{zero}f\left(x\right)=0,\mathrm{\forall }x\in \mathbb{R},\mathrm{so}$$$$f\left(x\right)=0=f(-x)\Rightarrow 0\in \mathrm{G}.\mathrm{G}\ne \varphi .$$$$\mathrm{G}\subseteq \mathrm{F}.$$$$\mathrm{let}\mathrm{be}f,g\in \mathrm{G}$$$$(f+g)\left(x\right)=f\left(x\right)+g\left(x\right)$$$$=-f(-x)+[-g(-x)]$$$$=-(f+g)(-x),\mathrm{\forall }x\in \mathbb{R}$$$$f+g\in \mathrm{G}.$$$$\mathrm{for}\alpha \in \mathbb{R},\left(\alpha f\right)\left(x\right)=\alpha (-f(-x))=-\left(\alpha f\right)(-x)$$$$\Rightarrow \alpha f\in \mathrm{G}.$$$$$$

Commented by mathocean1 last updated on 18/Apr/20

$$\mathrm{thank}\mathrm{you}\mathrm{sir}!$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com