Nann

2021-02-21

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

$\overrightarrow{T}(x,y)=(r\mathrm{cos}(\theta +\phi ),r\mathrm{sin}(\theta +\phi ))$

for some constants$\mathrm{\angle}\phi $

(f) Reflection: given a fixed vector$\overrightarrow{r}=(a,b),\overrightarrow{T}$ maps each point to its reflection with

respect to$\overrightarrow{r}\overrightarrow{T}(\overrightarrow{x})=\overrightarrow{x}-2{\overrightarrow{x}}_{r\perp}$

$=2{\overrightarrow{x}}_{r}-\overrightarrow{x}$

(d) Rotation: if

for some constants

(f) Reflection: given a fixed vector

respect to

pattererX

Skilled2021-02-22Added 95 answers

For proving the linear transformation, use that the following properties:

and

where alpha and beta are the scalars.

(d)Given that,

and

for some constant

Now showing in below T is linear transformation,

and

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,

Now showing in below T is linear transformation,

and

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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