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Math Help - Properties of Transpose Proof!

  1. #1
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    Lightbulb Properties of Transpose Proof!

    does anyone maybe have a link to prove the properties of the transpose?
    Transpose - Wikipedia, the free encyclopedia

    My teacher said we should know how to prove these...
    In the website link, # 1 , 2 , 3 , 4 , and (A^r)^T = (A^T)^r for all nonnegative integers.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by aeubz View Post
    does anyone maybe have a link to prove the properties of the transpose?
    Transpose - Wikipedia, the free encyclopedia

    My teacher said we should know how to prove these...
    In the website link, # 1 , 2 , 3 , 4 , and (A^r)^T = (A^T)^r for all nonnegative integers.
    Hello

    Some of these proofs aren't that bad

    \left(A^T\right)^T=A

    Let A=\left[a_{ij}\right], A^T=\left[a_{ji}\right] and A^T=C=\left[c_{ij}\right]

    Thus, c_{ij}^T=c_{ji}=a_{ji}^T=a_{ij}

    \mathbb{Q.E.D.}

    -------------------------------------------------------------------------

    \left(A+B\right)^T=A^T+B^T

    Let A=\left[a_{ij}\right], B=\left[b_{ij}\right] and A+B=C=\left[c_{ij}\right]

    Thus, c_{ij}^T=c_{ji}=a_{ji}+b_{ji}=a_{ij}^T+b_{ij}^T

    \mathbb{Q.E.D.}

    -------------------------------------------------------------------------

    \left(cA\right)^T=cA^T

    Let A=\left[a_{ij}\right] and cA=C=\left[c_{ij}\right]

    Thus, c_{ij}^T=c_{ji}=ca_{ji}=ca_{ij}^T

    \mathbb{Q.E.D.}

    -------------------------------------------------------------------------

    Maybe other can elaborate on these...because I think I made it too short and sweet

    --Chris
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  3. #3
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    Quote Originally Posted by aeubz View Post
    (A^r)^T = (A^T)^r for all nonnegative integers.
    You need to prove (AB)^T = B^T A^T.

    Then by induction (A_1...A_k)^T = A_k^T ... A_1^T.

    Therefore, (A^r)^T = (A...A)^T = (A^T) ... (A^T) = (A^T)^r.
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  4. #4
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    Quote Originally Posted by ThePerfectHacker View Post
    Then by induction (A_1...A_k)^T = A_k^T ... A_1^T.
    Hm, what if I need to prove this? By induction I have to assume this is true and then try to prove it's true for A_n where n=k+1? I have to do it right now....I thought I would need to treat (A_1...A_k) as a single matrix, correct? I either have to prove it this way or prove it for three, assuming it's true for two. But I wanted to use mathematical induction. The proof for two is given in the book, so I kinda extended the idea, treating this whole thing as a single (A_1...A_k) matrix. Or there is a different way to prove it?
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