# 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 overrightarrow{T}(x,y)=(r cos(theta+ varphi), r sin (theta+ varphi)) for some constants angle varphi (f) Reflection: given a fixed vector overrightarrow{r} = (a, b), overrightarrow{T} maps each point to its reflection with respect to overrightarrow{r} overrightarrow{T}(overrightarrow{x})=overrightarrow{x}-2overrightarrow{x}_{r perp} =2 overrightarrow{x}_{r}-overrightarrow{x}

Question
Transformation properties
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
$$\overrightarrow{T}(x,y)=(r \cos(\theta+ \varphi), r \sin (\theta+ \varphi))$$
for some constants $$\angle \varphi$$
(f) Reflection: given a fixed vector $$\overrightarrow{r} = (a, b), \overrightarrow{T}$$ maps each point to its reflection with
respect to $$\overrightarrow{r} \overrightarrow{T}(\overrightarrow{x})=\overrightarrow{x}-2\overrightarrow{x}_{r \perp}$$
$$=2 \overrightarrow{x}_{r}-\overrightarrow{x}$$

2021-02-22
For proving the linear transformation, use that the following properties:
$$T(\alpha x + \beta y) = \alpha T (x) + \beta T (y)$$
and $$T(\alpha x) = \alpha T(x)$$
where alpha and beta are the scalars.
(d)Given that,
$$x = r \cos \theta$$
$$y = r \sin \theta$$
and $$\overrightarrow{T}(x,y)=(r \cos(\theta+\varphi), r \sin(\theta+\varphi))$$
for some constant $$\varphi$$
Now showing in below T is linear transformation,
$$T(\alpha x_{1}(x,y)+\beta x_{2}(x,y))=[\alpha\{r_{1} \cos(\theta= \varphi), r_{1} \sin (\theta=\varphi)\}+\beta\{r_{2} \cos(\theta=\varphi), r_{2} \sin (\theta=\varphi\}]$$
$$= \alpha T (r_{1} \cos(\theta = \varphi), r_{1} \sin(\theta = \varphi)) + \beta T (r_{2} \cos(\theta = \varphi), r_{2} \sin(\theta = \varphi))$$
$$= \alpha T (x_{1} (x, y)) + \beta T (x_{2}(x, y))$$
and $$T(\alpha(x, y)) = {\alpha r \cos(\theta = \varphi), \alpha r \sin(\theta = \varphi)}$$
$$= \alpha (r \cos(\theta = \varphi), r \sin(\theta = \varphi))$$
$$= \alpha T (x, y)$$
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, $$\overrightarrow{T}(\overrightarrow{x})=\overrightarrow{x}-2 \overrightarrow{x} r$$
$$=2\overrightarrow{x}_{r}-\overrightarrow{x}$$
Now showing in below T is linear transformation,
$$T(\alpha x + \beta y) = (\alpha x + \beta y) - 2 (\alpha x + \beta y)_{\varphi} \(= \alpha x + \beta y - 2 \alpha x_{\varphi} - brta y_{\varphi}$$
$$=(\alpha x - 2 \alpha x_{\varphi}) + (\beta y - \beta y_{\varphi})$$
$$= \alpha (x - 2x_{\varphi}) + \beta (y - y_{\varphi})$$
$$= \alpha T (x) + \beta T (y)$$
and $$T (\alpha x) = \alpha x - 2(\alpha x)_{\varphi}$$
$$= \alpha x - 2 \alpha x_{\varphi}$$
$$= \alpha (x- 2x_{\varphi})$$
$$= \alpha T (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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