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Math Help - Linear Algebra matrix problem

  1. #1
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    Linear Algebra matrix problem

    let A be an m x n matrix. prove that the set b = (vertor b a member of R^m : A*vector x = vector b is consistant) is a subspace of R^m

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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by bugal402 View Post
    let A be an m x n matrix. prove that the set b = (vertor b a member of R^m : A*vector x = vector b is consistant) is a subspace of R^m

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    I don't know what this means. Let A be an m\times n matrix. Prove that the set b=\left\{\bold{b}\in\mathbb{R}^m:A*\bold{x}=\bold{  b}\text{ is consistent}\right\}?
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  3. #3
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    I think you mean "the set of all b such that Ax= b for some x in R^m" - that is known as the image of R^m and is also the "column space" since it is spanned by the columns of the matrix.

    To prove any subset is a subspace, it is sufficient to show that it is closed under addition and scalar multiplication- or, combining the two, that if u and v are in the subset and a and b are scalars, then au+ bv is also in the subset.

    If u and v are in the subset then there exist x and y in R^m such that Ax= u and Ay= v.

    Look at A(au+ bv).
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