$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 $H$.$

Question

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

Report

Answer 1

Answers can be viewed only if

The questioner was satisfied and accepted the answer, or
The answer was disputed, but the judge evaluated it as 100% correct.

View the answer

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 $M_{3 \times 3}$.

1 Attachment

Nirenberg

Nirenberg

Please see the attached file for a basis for H.
Nirenberg

Please consider offering higher amounts in future, otherwise your question may not get answered. Note that the questions are answered by highly trained Mathematicians and the offer should worth their time.

$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 $H$.$

Answer

Related Questions

Search