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Math Help - Stuck on Subspace proof question

  1. #1
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    Stuck on Subspace proof question

    Question:
    Let A and B be mxn matrices and let S be the set of all x in R^n such that Ax = Bx. Show that S is a subspace of R^n.

    My approach:
    So i know that there are 3 properties for H to be a subspace in R^n.
    1) the zero vector is in H
    2) for each u and v in H, the sum u+v is in H
    3) for each u in H and each scalar c, the vector cu is in H

    So im confused on how i would go abouts proving this?
    Would i take property 3 and multiply Ax=BX by a scalar?

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  2. #2
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    Quote Originally Posted by kensington View Post
    Question:
    Let A and B be mxn matrices and let S be the set of all x in R^n such that Ax = Bx. Show that S is a subspace of R^n.

    My approach:
    So i know that there are 3 properties for H to be a subspace in R^n.
    1) the zero vector is in H
    2) for each u and v in H, the sum u+v is in H
    3) for each u in H and each scalar c, the vector cu is in H

    So im confused on how i would go abouts proving this?
    Would i take property 3 and multiply Ax=BX by a scalar?

    Sounds good to me!
    1) What are A0 and B0? Are they equal? (0 is the 0 vector)
    2) Suppose x and y are such that Ax=Bx and Ay= By. What are A(x+y) and B(x+y)? Are they equal?
    3) Suppose x is such that Ax= By andck is a scalar. What are A(cx) and B(cx)? Are they equal?
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