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Problem 64: Determine whether it is a subspace of the given vector space $\V$
$V = M_2(\R)$, and S is the subset of all $2\times2$ symmetric matrices

Problem 64: Determine whether it is a subspace of the given vector space $\V$
$V = M_2(\R)$, and S is the subset of all $2\times2$ symmetric matrices.

Solution:
$S= \{A \in M_2(\R) : A$ is symmetric matrices$\}$
$S= \{A \in M_2(\R) : A^T = A\}$
$S \ne \varnothing$ ; since $0_2\in S$.
Closure under Addition: If $A,B \in S$,
then $(A + B)^T = A^T + B^T = A + B$, which shows that $A + B \in S$.
Closure under Scalar Multiplication: If $r \in R$ and $A \in S$,
then $(rA)^T = rA^T = rA$, which shows that $rA \in S$.
Consequently, $S$ is a subspace of $M_2(\R)$

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