Which of the following are linear transformations from RR2→RR2? (d) Rotation: if x=rcos⁡θ,y=rsin⁡θ, then T→(x,y)=(rcos⁡(θ+φ),rsin⁡(θ+φ)) for some constants ∠φ (f) Reflection: given a fixed vector r→=(a,b),T→ maps each point to its reflection with respect to r→T→(x→)=x→−2x→r⊥ =2x→r−x→

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Which of the following are linear transformations from RR2→RR2?  (d) Rotation: if x=rcos⁡θ,y=rsin⁡θ, then  T→(x,y)=(rcos⁡(θ+φ),rsin⁡(θ+φ))   for some constants ∠φ  (f) Reflection: given a fixed vector r→=(a,b),T→ maps each point to its reflection with  respect to r→T→(x→)=x→−2x→r⊥  =2x→r−x→

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For proving the linear transformation, use that the following properties: T(αx+βy)=αT(x)+βT(y) and T(αx)=αT(x) where alpha and beta are the scalars. (d)Given that, x=rcos⁡θ y=rsin⁡θ and T→(x,y)=(rcos⁡(θ+φ),rsin⁡(θ+φ)) for some constant φ Now showing in below T is linear transformation, T(αx1(x,y)+βx2(x,y))=[α{r1cos⁡(θ=φ),r1sin⁡(θ=φ)}+β{r2cos⁡(θ=φ),r2sin⁡(θ=φ}] =αT(r1cos⁡(θ=φ),r1sin⁡(θ=φ))+βT(r2cos⁡(θ=φ),r2sin⁡(θ=φ)) =αT(x1(x,y))+βT(x2(x,y)) and T(α(x,y))=αrcos⁡(θ=φ),αrsin⁡(θ=φ) =α(rcos⁡(θ=φ),rsin⁡(θ=φ)) =αT(x,y) Hence, rotation is linear transformation. (f) Given that a fixed vector r and T maps each point to its reflection with respect to vector r, T→(x→)=x→−2x→r =2x→r−x→ Now showing in below T is linear transformation, T(αx+βy)=(αx+βy)−2(αx+βy)φ =αx+βy−2αxφ−brtayφ =(αx−2αxφ)+(βy−βyφ) =α(x−2xφ)+β(y−yφ) =αT(x)+βT(y) and T(αx)=αx−2(αx)φ =αx−2αxφ =α(x−2xφ) =αT(x) Hence, reflection is linear transformation. Therefore, above satisfies both properties of linear transformation. Hence, both rotation and reflection are linear transformation.

Which of the following are linear transformations from RR^{2} rightarrow RR^{2} ? (d) Rotation: if x = r cos theta, y = r sin theta, then overrightarr

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