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I have task sound like:

Examine that $ W $ is a subspace of the vector space $M_{3x3} $

$W=({A:A^t=-A}) $

To check it from definition I have to check two conditions.

1) $ \vec u + \vec v \in W $

2) $ \alpha \cdot \vec u \in W $

1) $B, C $ are matrix 3x3 and $\in W $

$B^t+C^t =(B+C)^t = -A $ and i don't know what i can do next. Could anyone tell me how i can do this task?

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1 Answer 1

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First Condition:

Two matrices in the subspace are $B$ and $C$. These satisfy: $$B^t = -B$$ $$C^t = -C$$

You must show that the sum of the two matrices is also in the subspace:

$$B^t + C^t = -B - C$$ $$(B + C)^t = -(B+C)$$

Which is in the subspace.

Don't compare the matrices to $A$; $A$ is an arbitrary matrix in the subspace used to explain the definition of the subspace.


Second condition:

Let $B$ be in the subspace (that is, $B^t = -B$), and $a$ be an arbitrary scalar. $$(a \cdot B)^t = a\cdot B^t$$ Because $B^t = -B$: $$(a \cdot B)^t = a \cdot (-B)$$ $$(a \cdot B)^t = -(aB)$$

Thus $a \cdot B$ is in the subspace.

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Thanks, I understand now, but how should i go with second condition? $ \alpha B^t = -B $ , yes? –  Emil Dec 2 '12 at 14:08
Thank you very much, now it's all clear. I have 2 more subparts of this task, i'll write what I know in new post in this topic. –  Emil Dec 2 '12 at 14:45
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