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

Answer:[tex]\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][/tex]Step-by-step explanation:We are given that vector space of all symmetric matrixM(R)={[tex]\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right][/tex] where a,b,c,d,e,f,g,h,i[tex]\in R[/tex]}Let A=[tex]\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right][/tex]A'=[tex]\left[\begin{array}{ccc}a&d&g\\b&e&h\\c&f&i\end{array}\right][/tex]A is symmetric Then A'=A[tex]\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right][/tex]=[tex]\left[\begin{array}{ccc}a&d&g\\b&e&h\\c&f&i\end{array}\right][/tex]a=a  , e=e,  i=ia-a=0 , e-e=0,   i-i=0b=d,  c=g  ,f=hHence, the matrix [tex]\left[\begin{array}{ccc}0&b&c\\b&0&f\\c&f&0\end{array}\right][/tex]Therefore, the basis are [tex]\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][/tex]There are three elements  to generate an element of vectors of all symmetric matrix [tex]3\times 3[/tex] matrix[tex]\left[\begin{array}{ccc}0&b&c\\b&0&f\\c&f&0\end{array}\right][/tex]=[tex]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][/tex]