$Let $H$ be the subset of all 3x3 matrices that satisfy $A^T$ = $-A$. Carefully prove that $H$ is a subspace of $M_{3x3} $ . Then find a basis for $H$.$

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$Let $H$ be the subset of all 3x3 matrices that satisfy $A^T$ = $-A$. Carefully prove that $H$ is a subspace of $M_{3x3} $ . Then find a basis for $H$.$

I would not only like to know the answer, but how you tackled this problem. The solution should not be too complex or jargon-dominated, as this is for an intro-level LA class

Linear Algebra

Nicholas Feruch

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Let $A_1, A_2 \in H$. To show that $H$ is a subspace, we need to show that
\[\alpha A_1 +\beta A_2 \in H, \text{for all} \alpha,\beta \in \mathbb{R}.\]
We have
\[(\alpha A_1 +\beta A_2 )^T=\alpha A_1^T+\beta A_2^T=\alpha (-A_1)+\beta (-A_2)\]
\[=-(\alpha A_1 +\beta A_2).\]
Hence
\[(\alpha A_1 +\beta A_2 )^T=-(\alpha A_1 +\beta A_2),\]
which means $\alpha A_1 +\beta A_2 \in H$, $\alpha,\beta \in \mathbb{R}.$ So $H$ is a subspace of $M_{3 \times 3}$.

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Please see the attached file for a basis for H.
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$Let $H$ be the subset of all 3x3 matrices that satisfy $A^T$ = $-A$. Carefully prove that $H$ is a subspace of $M_{3x3} $ . Then find a basis for $H$.$

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