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squenshl

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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. Let's 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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