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Homework Help: Proving T is linear

  1. Nov 23, 2005 #1
    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...
     
  2. jcsd
  3. Nov 23, 2005 #2
    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
     
  4. Nov 23, 2005 #3
    sorry, i think it means the transpose... cos in the question its A to the power of capital T...
     
  5. Nov 23, 2005 #4

    Galileo

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    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?
     
  6. Nov 24, 2005 #5
    i dont understand what happens to the transpose though
     
  7. Nov 24, 2005 #6

    Galileo

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    Are there any rules or identities with the transpose you are familiar with?
     
  8. Nov 24, 2005 #7
    Use the fact (that you already know from your original post) that A^T is linear to prove that A + A^T is linear.
     
  9. Nov 24, 2005 #8
    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
     
  10. Nov 25, 2005 #9

    Galileo

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    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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