1. Let u=(u1,u2,u3) and v=(v1,v2,v3) be vectors in R3 and c and d be scalars.
Consider
T(cu+dv)=T(cu1+dv1,cu2+dv2,cu3+dv3)
=(cu1+dv1−(cu2+dv2)+cu3+dv3,
−2(cu1+dv1)+2(cu2+dv2)−2(cu3+dv3))
=(c(u1−u2+u3)+d(v1−v2+v3),
c(−2u1+2u2−2u3)+d(−2v1+2v2−2v3))
=c(u1−u2+u3,−2u1+2u2−2u3)
+d(v1−v2+v3,−2v1+2v2−2v3)
=cT(u)+dT(v) Therefore the transformation T:R3→R2, given by
T(x,y,z)=(x−y+z,−2x+2y−2z) is linear.
2.
T(e1)=T(1,0,0)
=(1−0+0,−2+0−0)=(1,−2)
T(e2)=T(0,1,0)
=(0−1+0,−0+2−0)=(−1,2)
T(e3)=T(0,0,1)
=(0−0+1,−0+0−2)=(1,−2) So
A=(1−2−121−2)
3.
T(0,0,0)=(0,0)
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