Question

Find the basis of the space Vof all symmetric 3X3 matrices, and thus determine the dimension of V.

Short answer

The dimension of a 3X3 symmetric matrices is 6 which is spanned by .

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

Step-by-step solution

Determine the basis.

Consider the matrix $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{12}& a_{22}& a_{23}\\ a_{13}& a_{23}& a_{33}\end{array}\right]$where all$a_{ij}$are real.

The general form of any symmetric matrix is $\left[\begin{array}{ccc}a& b& d\\ b& c& e\\ d& e& f\end{array}\right]$.

Simplify the equation $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{12}& a_{22}& a_{23}\\ a_{13}& a_{23}& a_{33}\end{array}\right]$as follows.

role="math" localid="1660127073394" $$\begin{gathered} A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{12}& a_{22}& a_{23}\\ a_{13}& a_{23}& a_{33}\end{array}\right] \\ A=\left[\begin{array}{ccc}{a}_{11}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]+\left[\begin{array}{ccc}0& a_{12}& 0\\ a_{12}& 0& 0\\ 0& 0& 0\end{array}\right]+\left[\begin{array}{ccc}0& 0& a_{13}\\ 0& 0& 0\\ a_{13}& 0& 0\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 0& a_{22}& 0\\ 0& 0& 0\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& a_{23}\\ 0& a_{23}& 0\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& a_{33}\end{array}\right] \\ A=a_{11}\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]+a_{12}\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]+a_{13}\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 1& 0& 0\end{array}\right]+a_{22}\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]+a_{23}\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]+a_{33}\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right] \end{gathered}$$

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

linear independent.

Therefore, the matrix A is spanned by .

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

Hence, the dimension of A is 6 and spanned by .

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