# Subspace of vector space.

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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### 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.

EDIT:

### 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