Let the trace function be Tr(M) = m1,1 + +mn,n. Define also the sum down

Question:

Let the trace function be Tr(M) = m1,1 + ∙∙ ∙ +mn,n. Define also the sum down the other diagonal Tr*(M) = m1,n + ∙∙ ∙+ mn,1.
(a) Show that the two functions Tr,Tr* : Mn×n → R are linear.
(b) Show that the function θ: Mn×n → R2 given by θ(M) = (Tr(M),Tr*(m)) is linear.
(c) Generalize the prior item.