Question

Find a basis for the vector space of all symmetric 3 x 3 matrices.

Answer 1

Answer:

$\left[\begin{array}{ccc}0&1&0\\1&0&0\\0&0&0\end{array}\right],\left[\begin{array}{ccc}0&1&0\\1&0&0\\0&0&0\end{array}\right],\left[\begin{array}{ccc}0&1&0\\1&0&1\\0&1&0\end{array}\right]$

Step-by-step explanation:

We are given that vector space of all symmetric matrix

M(R)={$\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right]$

where a,b,c,d,e,f,g,h,i$\in R$}

Let A=$\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right]$

A'=$\left[\begin{array}{ccc}a&d&g\\b&e&h\\c&f&i\end{array}\right]$

A is symmetric

Then A'=A

$\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right]$=$\left[\begin{array}{ccc}a&d&g\\b&e&h\\c&f&i\end{array}\right]$

a=a  , e=e,  i=i

a-a=0 , e-e=0,   i-i=0

b=d,  c=g  ,f=h

Hence, the matrix

$\left[\begin{array}{ccc}0&b&c\\b&0&f\\c&f&0\end{array}\right]$

Therefore, the basis are

$\left[\begin{array}{ccc}0&1&0\\1&0&0\\0&0&0\end{array}\right],\left[\begin{array}{ccc}0&1&0\\1&0&0\\0&0&0\end{array}\right],\left[\begin{array}{ccc}0&1&0\\1&0&1\\0&1&0\end{array}\right]$

There are three elements  to generate an element of vectors of all symmetric matrix $3\times 3$ matrix

$\left[\begin{array}{ccc}0&b&c\\b&0&f\\c&f&0\end{array}\right]$=$b\left[\begin{array}{ccc}0&1&0\\1&0&0\\0&0&0\end{array}\right]+c\left[\begin{array}{ccc}0&1&0\\1&0&0\\0&0&0\end{array}\right]+f\left[\begin{array}{ccc}0&1&0\\1&0&1\\0&1&0\end{array}\right]$