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Matrix subspace

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Please see the attached file thank-you for your help.

Let A be an n x n matrix, and suppose "T" is a subspace of . Define A(T) to be the set that results from multiplying "A" times vectors in "T" in all possible ways; that is, .
(a) Show that A(T) is also a subspace of .
(b) For the particular case that A =
And T = , which is a
subspace of , what do you get for the set A(T)? Describe the set; also, find a basis for it.

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This provides an example of working with a matrix to complete a proof regarding subspaces.

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Proof:
1. Since is a subspace in , then for any , , we ...

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