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Question

# Show that the set W of all symmetric $(3 \times 3)$ matrices is a subspace of the vector space of all $(3 \times 3)$ matrices. Find a spanning set for W.

Solution

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$\theta$ in the vector space of all $3\times3$ matrices is

$\begin{bmatrix} 0&0&0\\0&0&0\\0&0&0 \end{bmatrix}$

. Conclude that $\theta^t=\theta$ therefore $\theta\in W$.

Choose any $A, B\in W$. $(A+B)^t=A^t+B^t=A+B$ so $A+B\in W$.

Chhose any scalar $c$. $(cA)^t=A^tc^t=Ac=cA$ so $cA\in W$.

Conclude that $W$ is a subspace of the vector space of all $3\times3$ matrices.

Choose any $A\in W$. Then

$A=\begin{bmatrix} a_{11}&a_{12}&a_{13}\\a_{12}&a_{22}&a_{23}\\a_{13}&a_{23}&a_{33} \end{bmatrix}$

.

Conclude that a spanning set for $W$ is $\{B_1, B_2, ... , B_6\}$ where

$B_1=\begin{bmatrix} 1&0&0\\0&0&0\\0&0&0 \end{bmatrix}$

,

$B_2=\begin{bmatrix} 0&1&0\\1&0&0\\0&0&0 \end{bmatrix}$

,

$B_3=\begin{bmatrix} 0&0&1\\0&0&0\\1&0&0 \end{bmatrix}$

,

$B_4=\begin{bmatrix} 0&0&0\\0&1&0\\0&0&0 \end{bmatrix}$

,

$B_5=\begin{bmatrix} 0&0&0\\0&0&1\\0&1&0 \end{bmatrix}$

and

$B_6=\begin{bmatrix} 0&0&0\\0&0&0\\0&0&1 \end{bmatrix}$

.

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