Suppose T: V -> V\' and T\':V -> V\' are linear. Show that T+T\' : V-> V\' defined by (T+ T\')(v)= T(v) + T\'(v) is linear If V and V\' are infinite- dimensional, determine the matrix of T + T\' in terms of the matrices of T and T\'. Solution To show that A=T+Tl is linear, it is required to show, A(ax+y)=aA(x)+A(y) for every x,y in V and \'a\' a scalar. Let x,y in V be arbitrary and \'a\' be a scalar. Then A(ax+y)=(T+Tl)(ax+y)=T(ax+y)+Tl(ax+y) (by definition of T+Tl) =aT(x)+T(y)+aT1(x)+Tl(y) (Since T and Tl are linear) =a(T(x)+T1(x))+(T(y)+Tl(y)) =a(T+Tl)(x)+(T+Tl)(y) =aA(x)+A(y) Hence A is a linear map. Now let matrix of T be B=(bij), that is the (i,j)th entry of the matrix is bij. That of Tl be C=(cij) Since for every x in V, (T+Tl)(x)=T(x)+Tl(x), while calculating the matrix of , we can observe that the corresponding entries of the matrices of T and Tl gets added up. Hence the matrix of T+Tl will be (bij+cij), that is matrix of T+Tl will be B+C, To be more clear, matrix of T+Tl will be the sum of matrices of T and Tl..