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Math Help - Linear algebra. A proof

  1. #1
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    Linear algebra. A proof

    Hi

    I want to prove
    (ab)AB = (aA)(bB), in which a, b E R and matrix multiplication AB are defined.

    Proof.
    Let A =(r_ij), B = (s_ij)

    (aA + bB)
    = ((a * r_ij) + (b * s_ij))
    = (a * r_ij + b * s_ij)
    = (a * r_ji + b * s_ji)
    = (a * r_ji) + (b * s_ji)
    = aA + bB


    But I don't know are there errors in a proof. ?
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  2. #2
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    Re: Linear algebra. A proof

    I think you are working at this too hard. We know that m(AB) = (mA)B = A(mB) since scalar multiplication commutes with the A and B operations. So you have
    (ab)(AB) = (abA)B = (aAb)B = (aA)(bB)

    -Dan
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  3. #3
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    Re: Linear algebra. A proof

    Thank you. I see, and a proof is that short? Can you explain it more, so I can understand that proof, please.
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    Forum Admin topsquark's Avatar
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    Re: Linear algebra. A proof

    Quote Originally Posted by ThinkingNumbers24 View Post
    Thank you. I see, and a proof is that short? Can you explain it more, so I can understand that proof, please.
    No real proof is needed. By definition we know that, for a scalar n and an operator N that scalar multiplication is defined as
    nN = Nn...Any operator commutes with a scalar. So that gives (ab)AB = (abA)B. And the rest is simply moving b to the other side of A.

    It occurs to me that you might actually have to prove aA = Aa from another list of assumptions?

    -Dan
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  5. #5
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    Re: Linear algebra. A proof

    Thank you again. I just have that I want to prove
    (ab)AB = (aA)(bB), in which a, b E R and matrix multiplication AB are defined.
    I thought that above can be proven without by definitions.

    Well, indeed, I think more, I believe that I will understand your answers.
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