Prove that T(x,y)=(3x+y,2y,x-y) defines a linear transformation T:R^2 -> R^3.

banganX 2020-11-07 Answered

Prove that T(x,y)=(3x+y,2y,xy) defines a linear transformation T:R2R3. Give the full and correct answer.

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Expert Answer

StrycharzT
Answered 2020-11-08 Author has 102 answers
Let (x1,y1),(x2,x2)R2andtc1,c2R. To prove that T is a linear transformation, we must prove that
T(c1(x1,y1)+c2(x2,y2))=c1T(x1,y1)+c2T(x2,y2)
This is done by a direct computation:
T(c1(x1,y1)+c2(x2,y2))
=T(c1x1,c2x2)+(c2x2,c2x2)
=T(c1x1c2x2,c1y1+c2y2)
=(3(c1x1+c2x2)+(c1y1+c2y2),2(c1y1+c2y2),(c1x1+c2x2)(c1y1+c2y2))
=(3c1x1+3c2x2+c1y1+c2y2,2c1y1+c2y2,c1x1+c2x2c1y1+c2y2)
=(3c1x1+c1y1,2c1y
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