allhvasstH
2021-01-08
Answered

Whether the function is a linear transformation or not.

$T\text{}:\text{}{R}^{2}\to {R}^{2},T(x,y)=(x,{y}^{2})$

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okomgcae

Answered 2021-01-09
Author has **93** answers

Calculation:

The function is defined as,

$T(x,y)=(x,{y}^{2})$

Assume two general vectors$u=({u}_{1},{u}_{2})$ and $v=({v}_{1},{v}_{2})$

Then$u+v=({u}_{1}+{v}_{1},{u}_{2}+{v}_{2})$

$cu=(c{u}_{1},c{u}_{2})$

The function is a linear transformation if it satisfies the two properties as mentioned in the approach part.

Compute$T(u+v)$ and $T\left(u\right)+T\left(v\right)$ as

$T(u+v)=T({u}_{1}+{v}_{1},\text{}{u}_{2}+{v}_{2})$

$=({u}_{1}+{v}_{1},\text{}{({u}_{2}+{v}_{2})}^{2})$

$=({u}_{1}+{v}_{1},\text{}{u}_{2}^{2}+{v}_{2}^{2}\text{}+\text{}2{u}_{2}{v}_{2})$

$T\left(u\right)+T\left(v\right)=T({u}_{1},\text{}{u}_{2})+T({v}_{1}\text{},{v}_{2})$

$=({u}_{1},\text{}{u}_{2}^{2})+({v}_{1},\text{}{v}_{2}^{2})$

$=({u}_{1}+{v}_{1},\text{}{u}_{2}^{2}+{v}_{2}^{2})$

Since$T(u+v)\ne qT\left(u\right)+T\left(v\right)$ , the first property is not satisfied.

Therefore, the function is not a linear transformation.

The function is defined as,

Assume two general vectors

Then

The function is a linear transformation if it satisfies the two properties as mentioned in the approach part.

Compute

Since

Therefore, the function is not a linear transformation.

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