# 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

Which of the following are linear transformations from $R{R}^{2}\to R{R}^{2}?$
(d) Rotation: if $x=r\mathrm{cos}\theta ,y=r\mathrm{sin}\theta ,$ then
$\stackrel{\to }{T}\left(x,y\right)=\left(r\mathrm{cos}\left(\theta +\phi \right),r\mathrm{sin}\left(\theta +\phi \right)\right)$
for some constants $\mathrm{\angle }\phi$
(f) Reflection: given a fixed vector $\stackrel{\to }{r}=\left(a,b\right),\stackrel{\to }{T}$ maps each point to its reflection with
respect to $\stackrel{\to }{r}\stackrel{\to }{T}\left(\stackrel{\to }{x}\right)=\stackrel{\to }{x}-2{\stackrel{\to }{x}}_{r\perp }$
$=2{\stackrel{\to }{x}}_{r}-\stackrel{\to }{x}$
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pattererX

For proving the linear transformation, use that the following properties:
$T\left(\alpha x+\beta y\right)=\alpha T\left(x\right)+\beta T\left(y\right)$
and $T\left(\alpha x\right)=\alpha T\left(x\right)$
where alpha and beta are the scalars.
(d)Given that,
$x=r\mathrm{cos}\theta$
$y=r\mathrm{sin}\theta$
and $\stackrel{\to }{T}\left(x,y\right)=\left(r\mathrm{cos}\left(\theta +\phi \right),r\mathrm{sin}\left(\theta +\phi \right)\right)$
for some constant $\phi$
Now showing in below T is linear transformation,
$T\left(\alpha {x}_{1}\left(x,y\right)+\beta {x}_{2}\left(x,y\right)\right)=\left[\alpha \left\{{r}_{1}\mathrm{cos}\left(\theta =\phi \right),{r}_{1}\mathrm{sin}\left(\theta =\phi \right)\right\}+\beta \left\{{r}_{2}\mathrm{cos}\left(\theta =\phi \right),{r}_{2}\mathrm{sin}\left(\theta =\phi \right\}\right]$
$=\alpha T\left({r}_{1}\mathrm{cos}\left(\theta =\phi \right),{r}_{1}\mathrm{sin}\left(\theta =\phi \right)\right)+\beta T\left({r}_{2}\mathrm{cos}\left(\theta =\phi \right),{r}_{2}\mathrm{sin}\left(\theta =\phi \right)\right)$
$=\alpha T\left({x}_{1}\left(x,y\right)\right)+\beta T\left({x}_{2}\left(x,y\right)\right)$
and $T\left(\alpha \left(x,y\right)\right)=\alpha r\mathrm{cos}\left(\theta =\phi \right),\alpha r\mathrm{sin}\left(\theta =\phi \right)$
$=\alpha \left(r\mathrm{cos}\left(\theta =\phi \right),r\mathrm{sin}\left(\theta =\phi \right)\right)$
$=\alpha T\left(x,y\right)$
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, $\stackrel{\to }{T}\left(\stackrel{\to }{x}\right)=\stackrel{\to }{x}-2\stackrel{\to }{x}r$
$=2{\stackrel{\to }{x}}_{r}-\stackrel{\to }{x}$
Now showing in below T is linear transformation,
$T\left(\alpha x+\beta y\right)=\left(\alpha x+\beta y\right)-2\left(\alpha x+\beta y{\right)}_{\phi }$
$=\alpha x+\beta y-2\alpha {x}_{\phi }-brta{y}_{\phi }$
$=\left(\alpha x-2\alpha {x}_{\phi }\right)+\left(\beta y-\beta {y}_{\phi }\right)$
$=\alpha \left(x-2{x}_{\phi }\right)+\beta \left(y-{y}_{\phi }\right)$
$=\alpha T\left(x\right)+\beta T\left(y\right)$
and $T\left(\alpha x\right)=\alpha x-2\left(\alpha x{\right)}_{\phi }$
$=\alpha x-2\alpha {x}_{\phi }$
$=\alpha \left(x-2{x}_{\phi }\right)$
$=\alpha T\left(x\right)$
Hence, reflection is linear transformation.
Therefore, above satisfies both properties of linear transformation.
Hence, both rotation and reflection are linear transformation.