T: R 3 --------> R 3 such that forevery v = (x,y,z) it is a member of R 3 T( v )
ID: 2939553 • Letter: T
Question
T: R3--------> R3 such that forevery v = (x,y,z) it is a member of R3T(v) = T((x,y,z)) = (y,y-z,z)
How do you show that T is a linear transformation?
Explanation / Answer
Question Details: T: R3-------->R3 such that for every v = (x,y,z) it is amember of R3 T(v) = T((x,y,z)) = (y,y-z,z) How do you show that T is a linear transformation? LET US DESIGNATE T[V]=U LET V1=[X1,Y1,Z1] AND V2=[X2,Y2,Z2] T[V1]=T[X1,Y1,Z1]=[Y1,Y1-Z1,Z1] T[V2]=T[X2,Y2,Z2]=[Y2,Y2-Z2,Z2] LET A BE ANY SCALAR LET US CHECK IF 1.T[V1+V2]=T[V1]+T[V2] LHS=T[(X1+X2),(Y1+Y2),(Z1+Z2)]=[(Y1+Y2),(Y1+Y2-Z1-Z2),(Z1+Z2)] RHS=[Y1,Y1-Z1,Z1]+[Y2,Y2-Z2,Z2]=[(Y1+Y2),(Y1+Y2-Z1-Z2),(Z1+Z2)]=LHS SO T[V1+V2]=T[V1]+T[V2].....OK 2.T[AV1]=A*T[V1] LHS=T[AX1,AY1,AZ1]=[AY1,AY1-AZ1,AZ1] RHS=A*T[V1]=A*[Y1,Y1-Z1,Z1]=[AY1,AY1-AZ1,AZ1]=LHS SO T[AV1]=A*T[V1].....OK HENCE T IS A LINEAR TRANSFORM
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