Proving T is linear

Another question that i'm a little bit stumped on...

Define T: Mnn -> Mnn by T(A) = A + A^T. Prove that T is linear

i know that with T(A) = A^T you can prove it by the equations (if A and B are arbitrary matrices in Mnn and c is a scalar):
T(A+B) = (A+B)^T = (A)^T + (B)^T = T(A) + T(B)
T(cA) = (cA)^T = cT(A)

but i dont know how to manipulate these equations for the A + A^T problem...
do you mean T(A)=A+A^T(A) ? (there's a redundant symbol describing not the same things) or T(A)=A+transpose(A)....the first is almost surely not linear, the second should be accepted to be proven like : T(A+B)=A+B+trn(A+B)=A+B+trn(A)+trn(B)=A+trn(A)+B+trn(B)=T(A)+T(B) with your defintion...the same for scalar multiplication
sorry, i think it means the transpose... cos in the question its A to the power of capital T...


Science Advisor
Homework Helper
You have to show that T(A+B)=T(A)+T(B) and T(cA)=cT(A)
Since T(A)=A+A^T, what do you get if you let T act on A+B?
i dont understand what happens to the transpose though


Science Advisor
Homework Helper
Are there any rules or identities with the transpose you are familiar with?
Use the fact (that you already know from your original post) that A^T is linear to prove that A + A^T is linear.
the transpose rules i know of are (A+B)transpose = Atranspose + Btranspose; (cA)transpose = cAtranspose; (AB)transpose = BtransposeAtranspose; and (Atranspose)transpose = A

but i still dont understand how to go about manipulating the addition and scalar multiplication formulas to prove that T(A)=A+Atranspose is linear


Science Advisor
Homework Helper
You know what T does to an arbitrary matrix Q right? T(Q)=Q+Q^T.
So what you have to show is that for any two matrices A,B we have T(A+B)=T(A)+T(B)

So let Q=A+B. Then T(A+B)=(A+B)+(A+B)^T.
We also have T(cA)=(cA)+(cA)^T.

So are these equal to T(A)+T(B) and cT(A) respectively?

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