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 Homework Statement:

1. Let ##S## be a subspace of ##\mathbb{R}^m## and let ##A## be a ##m\times n## matrix.
Prove that the set ##T:= \left\{\mathbf{x}\in \mathbb{R}^n:A\mathbf{x}\in S\right\}## is a subspace of ##\mathbb{R}^n##.
 Relevant Equations:
 None
1. Lets show the three conditions for a subspace are satisfied:
Since ##\mathbf{0}\in \mathbb{R}^n##, ##A\times \mathbf{0} = \mathbf{0}\in S##.
Suppose ##x_1, x_2\in \mathbb{R}^n##, then ##A(x_1+x_2) = A(x_1)+A(x_2)\in S##.
Suppose ##x\in S## and ##\lambda\in \mathbb{R}##, then ##A(\lambda x) = \lambda A(x)\in S##.
Is this correct???
Since ##\mathbf{0}\in \mathbb{R}^n##, ##A\times \mathbf{0} = \mathbf{0}\in S##.
Suppose ##x_1, x_2\in \mathbb{R}^n##, then ##A(x_1+x_2) = A(x_1)+A(x_2)\in S##.
Suppose ##x\in S## and ##\lambda\in \mathbb{R}##, then ##A(\lambda x) = \lambda A(x)\in S##.
Is this correct???
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