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Question Number 142236 by mathocean1 last updated on 28/May/21

$$\mathrm{f}\mathrm{is}\mathrm{an}\mathrm{endomorphism}\mathrm{of}\mathrm{V}\mathrm{such}$$$$\mathrm{that}\mathrm{f}\circ \mathrm{f}=-{\mathrm{Id}}_{\mathrm{V}}.$$$$1.\mathrm{Show}\mathrm{that}\mathrm{f}\mathrm{is}\mathrm{an}\mathrm{isomorphism}\mathrm{of}$$$$\mathrm{V}\mathrm{and}\mathrm{express}{\mathrm{f}}^{-1}\mathrm{in}\mathrm{function}\mathrm{of}\mathrm{f}.$$$$2.\mathrm{show}\mathrm{that}\overrightarrow{0}\mathrm{is}\mathrm{the}\mathrm{one}\mathrm{invariant}$$$$\mathrm{vector}\mathrm{by}\mathrm{f}.$$$$3.\mathrm{Given}\overrightarrow{\mathrm{u}}\ne \overrightarrow{0}\mathrm{and}\overrightarrow{\mathrm{u}}\in \mathrm{V}.$$$$\mathrm{a}.\mathrm{Show}\mathrm{that}(\overrightarrow{\mathrm{u}};\mathrm{f}\left(\overrightarrow{\mathrm{u}}\right))\mathrm{is}\mathrm{a}\mathrm{base}\mathrm{of}\mathrm{V}.$$$$\mathrm{b}.\mathrm{Write}\mathrm{the}\mathrm{matrix}\mathrm{of}\mathrm{f}\mathrm{in}\mathrm{base}$$$$(\overrightarrow{\mathrm{u}};\mathrm{f}\left(\overrightarrow{\mathrm{u}}\right)).$$

Answered by mindispower last updated on 28/May/21

$$fo(-f)=Idv=(-f)o\left(f\right)=idv$$$$f^{-}=-f$$$$2letuinvariant\Rightarrow f\left(u\right)=u\Rightarrow fof\left(u\right)=f\left(u\right)=u$$$$\Rightarrow -u=u\Rightarrow 2u=0\Rightarrow u=\overrightarrow{0}$$$$leta,b\in {\mathbb{R}}^2\mid au+bf\left(u\right)=0\dots .\times a$$$$af\left(u\right)-bu=0\dots .\times -b$$$$\Rightarrow (b^2+a^2)u=0\Rightarrow a=b=0linearindependenent$$$$generatricehavewedim\left(V\right)=2\dots ?$$

Commented by mathocean1 last updated on 28/May/21

$$thankyousir.$$$$yesdim\left(V\right)=2.$$$$whatcouldbethematrix..$$

Commented by mindispower last updated on 28/May/21

$$\left(\begin{array}{c}0-1\\ 10\end{array}\right)$$

Commented by mathocean1 last updated on 29/May/21

$$Thanks$$

Commented by mindispower last updated on 29/May/21

$$pleasur$$

Answered by mindispower last updated on 28/May/21

$$isomoprhismeisbijectionof[morphisme$$$$let(u,v)\in V^2\mid f\left(u\right)=f\left(v\right)\Rightarrow fof\left(u\right)=fof\left(v\right)\Rightarrow -u=-v$$$$\Rightarrow u=v\Rightarrow finjective$$$$letu\in V,wehavefof\left(u\right)=-u$$$$\Rightarrow fof(-u)=u\Rightarrow f\left(f(-u)\right)=uletY=f(-u)$$$$\Rightarrow \mathrm{\forall }u\in V\mathrm{\exists }y=f(-u)\mid f\left(y\right)=u\Rightarrow fsurjective$$$$sofismorphisme$$

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