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Math Help - Proving that something is a subspace

  1. #1
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    Proving that something is a subspace

    The question:

    Show that the set S = { b \in \mathbb{R}^2 : b = Ax \textrm{ for some } x \in \mathbb{R}^3} where A = \[ \left( \begin{array}{ccc}<br />
2 & -3 & 1 \\<br />
4 & 5 & -3 \end{array} \right)\] is a subspace of \mathbb{R}^2. Explain why each column of the matrix belongs to the set S.

    I'm having trouble with this one. I recognize the Ax=b form, but I'm not sure how to prove that it is a subspace. I know the Subspace Theorem, and I managed to show that it isn't an empty-set by substituting x = 0 into the matrix. But when it comes to closure under vector addition, I'm getting confused. Could someone shed some light on how I should attempt this? Thanks.
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  2. #2
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    Let b_1,b_2 \in S so b_1+b_2=Ax_1 + Ax_2 = A(x_1+x_2) so that x_1+x_2 \in \mathbb{R}^3.
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