T(1,0,0)=(0,1,1), T(0,1,0)=(1,1,2) and T(0,0,1)=(1,2,3). Find T(2,2,-2). Please

Question

T(1,0,0)=(0,1,1), T(0,1,0)=(1,1,2) and T(0,0,1)=(1,2,3).

Find T(2,2,-2).

Please show all working out, especially how you derive what the transform actually is. It does not need to be in matrix form (actually we have not reached that yet) so please do not answer in the form of a matrix.

Explanation / Answer

since it is a linear transformation for any scalars a1,a2,a3,a4,not all zero and by definition of linear transformation, we have.. T(2,2,-2)=T(a1(1,0,0)+a2(0,1,0)+a3(0,0,1)) that is, =>(2,2,-2)=a1(1,0,0)+a2(0,1,0)+a3(0,0,1) =>(2,2,-2)=(a1,0,0)+(0,a2,0)+(0,0,a3) =>(2,2,-2)=(a1,a2,a3) therefore, a1=a2=2 and a3=-2 ;by equating both sides now required transformation T(2,2,-2)=T(a1(1,0,0)+a2(0,1,0)+a3(0,0,1)) =a1T(1,0,0)+a2T(0,1,0)+a3T(0,0,1) ;since,T(a(p,q)+b(x,y))=aT(p,q)+bT(x,y) =2(0,1,1)+2(1,1,2)-2(1,2,3) ;putting the values of a1,a2,a3 and the values of T(1,0,0),T(0,1,0),T(0,0,1) .... given in the question) =(0,0,3) (ans)

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