Suppose that U V and W are vector spaces and that R U right

Suppose that U, V, and W are vector spaces and that R : U right arrow V, S : U right arrow V, and T : V right arrow W are linear transformations and c is a scalar. Prove the following: R + S is linear. R - S is linear. cR is linear. T o R is linear.

Solution

(a)

(R+S)(u+v)=R(u+v)+S(u+v)=R(u)+R(v)+S(u)+S(v)=(R+S)(u)+(R+S)(v)

c(R+S)(u)=cR(u)+cS(v)=R(cu)+S(cv)=(R+S)(cv)

Hence R+S is linear.

(b)

(R-S)(u+v)=R(u+v)-S(u+v)=R(u)+R(v)-S(u)-S(v)=(R-S)(u)+(R-S)(v)

c(R-S)(u)=cR(u)-cS(v)=R(cu)-S(cv)=(R-S)(cv)

Hence R-S is linear.

c)

cR(u+v)=c(R(u)+R(v))=cR(u)+cR(v)

caR(u)=cR(au)

Hence ,cR is linear

d)

ToR(u+v)=T(R(u)+R(v))=ToR(u)+ToR(v)

cToR(u)=cT(R(u))=T(cR(u))=T(R(cu))=ToR(cu)

Hence, ToR is linear