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This was a test question that I got incorrect. I didn't like the way my teacher proved this afterwards, they said it IS a subspace of Mnxn. Any help in explaining how it could be would be greatly appreciated.

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- #1

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This was a test question that I got incorrect. I didn't like the way my teacher proved this afterwards, they said it IS a subspace of Mnxn. Any help in explaining how it could be would be greatly appreciated.

- #2

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Suppose B1 and B2 are matrices in H. We wish to show that (B1+B2) is also a matrix in H. To do so, we must verify that A(B1+B2) = (B1+B2)A. Let's start with the left hand side:

A(B1+B2) = A B1 + A B2 = B1 A + B2 A

(here we can replace A B1 with B1 A because B1 is an element of H, and hence satisfies AB1 = B1A; similarly for A B2).

Finally,

B1 A + B2 A = (B1+B2) A.

Hence, (B1+B2) satisfies the condition A(B1+B2) = (B1+B2)A. Therefore, (B1+B2) is also in H, so H is closed under addition. You finish the rest by showing H is closed under multiplication.

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JasonRox

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H is constructed from elements of M, so clearly this is not necessary at all.First, you need to show that H is contained within M (obvious; no work needed but for rigor, you should at least state that H is contained within M).

The only necessity to showing something is subspace is to show that it is non-empty, closed under addition and scalar multiplication.

What's the easiest way to show something is non-empty? Well, is the identity matrix in there?

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