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Math Help - Row vectors of B and the subspace spanned by row vectors of B lie in row space A:Why?

  1. #1
    Senior Member x3bnm's Avatar
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    Row vectors of B and the subspace spanned by row vectors of B lie in row space A:Why?

    I've some questions about the following theorem. The theorem is:

    If an m x n matrix A is row equivalent to an m x n matrix B, then the row space of A is equal to the row space of B.

    The proof is like this:

    Because the rows of B can be obtained from the rows of A by elementary row operations (scalar multiplication and addition), it follows that the row vectors of B can be written as linear combinations of the row vectors of A.

    The row vectors of B lie in the row space of A, and the subspace spanned by the row vectors of B is contained in the row space of A. (<-----My question why's that the row vectors of B lie in the row space of A, and why's the subspace spanned by the row vectors of B is contained in the row space of A?)

    But it is also true that the rows of A can be obtained from the rows of B by elementary row operations. So, you can conclude that the two row spaces are subspaces of each other, making them equal.

    -----------End of proof---------

    Can anyone kindly answer the marked question in above proof?

    If the row vecors of B can be written as linear combination of row vectors of A how it makes the statement "The row vectors of B lie in the row space of A, and the subspace spanned by the row vectors of B is contained in the row space of A" true?

    Is it possible to elaborate on this?
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  2. #2
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    Re: Row vectors of B and the subspace spanned by row vectors of B lie in row space A:

    the row space of A IS linear combinations of the rows of A. we got every row-vector of B by adding together scalar multiples of rows of A.

    that is precisely how you form linear combinations of vectors.
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  3. #3
    Senior Member x3bnm's Avatar
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    Re: Row vectors of B and the subspace spanned by row vectors of B lie in row space A:

    Quote Originally Posted by Deveno View Post
    the row space of A IS linear combinations of the rows of A. we got every row-vector of B by adding together scalar multiples of rows of A.

    that is precisely how you form linear combinations of vectors.
    Thanks a thousand time Deveno. You showed me the way again. Thank you for that.
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