determine whether T is a linear transformation T Mnn Mnn def

determine whether T is a linear transformation T: Mnn-> Mnn defined by T(A)=AB+BA, where B is a fixed nXn matrix. ( Linear Algebra problem)

Solution

yes T is linear mapping

T(A1 + cA2) = (A1 + cA2)B + B(A1 + cA2)

we know that

(A1 + cA2)B + B(A1 + cA2) = A1B + cA2B + BA1 + cBA2 = (A1B + BA1) + c(A2B + BA2)

but this last expression is equal to T(A1) + cT(A2)