Whether the function is a linear transformation or not. T : R^{2} rightarrow R^{3}, T(x,y)=(sqrt{x},xy,sqrt{y})

alesterp 2021-03-06 Answered
Whether the function is a linear transformation or not.
T : R2R3,T(x,y)=(x,xy,y)
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Expert Answer

Aubree Mcintyre
Answered 2021-03-07 Author has 73 answers

Calculation:
The function is defined as,
T(x,y)=(x,xy,y)
Assume two general vectors u=(u1,u2) and v=(v1,v2)
Then u+v=(u1+v1,u2+v2)
cu=(cu1,cu2)
The function is a linear transformation if it satisfies the two properties as mentioned in the approach part.
Compute T(u+v) and T(u)+T(v) as,
T(u+v)=T(u1+v1,u2+v2)
=(u1+v1,(u1+v1)(u2+v2),u2+v2)
=(u1+v1,(u1u2+u1y2+u2v1+v1v2),(u2+v2)
T(u)+T(v)=T(u1,u2)+T(v1,v2)
=(u1,u1u2,u2)+(v1,v1v2,v2)
=(u1+v1,u1u2+v1v2,u2+v2
Since T(u+v)qT(u)+T(v), the first property is not satisfied.
Therefore, the function is not a linear transformation.

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