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}

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
asked 2021-02-21
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 \(\overrightarrow{r} \overrightarrow{T}(\overrightarrow{x})=\overrightarrow{x}-2\overrightarrow{x}_{r \perp}\)
\(=2 \overrightarrow{x}_{r}-\overrightarrow{x}\)

Answers (1)

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.
0

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