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

Nann 2021-02-21 Answered
Which of the following are linear transformations from RR2RR2?
(d) Rotation: if x=rcosθ,y=rsinθ, 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 rT(x)=x2xr
=2xrx
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Expert Answer

pattererX
Answered 2021-02-22 Author has 95 answers

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=rcosθ
y=rsinθ
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)=x2xr
=2xrx
Now showing in below T is linear transformation,
T(αx+βy)=(αx+βy)2(αx+βy)φ
=αx+βy2αxφbrtayφ
=(αx2αxφ)+(βyβyφ)
=α(x2xφ)+β(yyφ)
=αT(x)+βT(y)
and T(αx)=αx2(αx)φ
=αx2αxφ
=α(x2xφ)
=αT(x)
Hence, reflection is linear transformation.
Therefore, above satisfies both properties of linear transformation.
Hence, both rotation and reflection are linear transformation.

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