Let S and T be linear transformations from V into W. Show that S + T and kT are both linear transformations, where (S + T)(v) = S(v) + T(v) and (kT)(v) = kT(v).

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Mitchel Aguirre · Accepted Answer

Step 1  Given   Let S and T be linear transformations from V into W.  where (S + T)(v) = S(v) + T(v) and (kT)(v) = kT(v).  To show : S + T is a linear transformation.  For any matrix X and Y in V and for any scalar c  (S+T)(X+Y)=S(X+Y)4+T(X +Y)  =S(X)+S(Y)+7T(X)+T(Y)  =S(X)+7T(X)+S(Y)+T(Y)  =(S+T)(X)+(S+T)(Y)  and  (S+T)(cX)=S(cX)+T(cX)  =cS(X)+cT(X)  =c[S(X)+T(X)]  =c(S+T)(x)  ⇒ S+T isa linear transformation. Step 3  To show : kT isa linear transformation.  For any matrix X and Y in V and for any scalar c  (kT)(X +Y)=T[k(X +Y)]  =T(kX +kY)  =T(kX)+T(kY)  =kT(X)+kT(Y)  (kT)(X) +(kT)(Y)  and  (kT)(cX)=kT(cT)=k[cT(X)]=c[kT(X)]=c(kT)(X)  ⇒ kT is a linear transformation.

Let S and T be linear transformations from V into W. Show that S + T and kT are

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