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Math Help - Proof of a very basic identity. a=b --> xa = xb

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    Proof of a very basic identity. a=b --> xa = xb

    Hi, I need to prove a very basic equality which states that
    if a=b , then x*a = x*b
    note that a,b, and x can be scalars, matrices or vectors.

    I guess I have to prove it using the ring (R, +, *), any help would be appreciated. Thanks!
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    Re: Proof of a very basic identity. a=b --> xa = xb

    Quote Originally Posted by publicvoid View Post
    I need to prove a very basic equality which states that
    if a=b , then x*a = x*b
    This is true for any theory with equality and a binary operation *. That is, this is a corollary of equality axioms as opposed to axioms describing any particular theory, such as theory of groups, rings or vector fields. Equality axioms are usually considered the common basis of all theories, along with such axioms as "A and (A -> B) imply B."

    Quote Originally Posted by publicvoid View Post
    note that a,b, and x can be scalars, matrices or vectors.

    I guess I have to prove it using the ring (R, +, *)
    Note that scalars (e.g., real numbers) and matrices do indeed form a ring, but vectors do not because of multiplication. The dot product maps two vectors into a number, not a vector, and the cross product is not associative, nor does it have a multiplicative identity. Vectors are usually considered to be a vector space over a field or its generalization, a module over a ring. In this case, multiplication takes two objects of different type: a scalar and a vector.
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    Re: Proof of a very basic identity. a=b --> xa = xb

    Hi, emakarov.
    Thanks for your answer.
    As I understand, you are saying that this is an axiom, which needs no proof. That is exactly what I think. But apparently my professor does not think so, so he gave it as a homework
    So, any ideas how to prove it?
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    Re: Proof of a very basic identity. a=b --> xa = xb

    I don't know how to prove it except for the following. Equality axioms may differ from source to source, but one approach is as follows. There are two axioms about equality: (1) for all x, x = x and (2) for all x, y, A[x] and x = y imply A[y] for any formula A. So, we have x * a = x * a by (1) and then replace one occurrence of a with b by (2) to get x * a = x * b.
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    Re: Proof of a very basic identity. a=b --> xa = xb

    Quote Originally Posted by publicvoid View Post
    . But apparently my professor does not think so, so he gave it as a homework . So, any ideas how to prove it?


    Without knowing the exact set of ring axioms, this is a mere guess.
    \begin{align*}0&=a-b\\ 0&=x(a-b)\\0 &=xa-xb\\0 &= ax-bx\\ax&=bx  \end{align*}
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    Re: Proof of a very basic identity. a=b --> xa = xb

    Quote Originally Posted by Plato View Post
    Without knowing the exact set of ring axioms, this is a mere guess.
    \begin{align*}0&=a-b\\ 0&=x(a-b)\\0 &=xa-xb\\0 &= ax-bx\\ax&=bx  \end{align*}
    yes this is how I did it, I hope it's true
    thanks for all answers.
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