Question

Which of the following are not a basis for the vector space of all symmetric \(2 \times 2\) matrices? Why?

 

a) \(\left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right),\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right),\left(\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right)\)
b) \(\left(\begin{array}{ll}3 & 3 \\ 3 & 3\end{array}\right),\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right),\left(\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right)\)
c) \(\left(\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right),\left(\begin{array}{cc}1 & 2 \\ 2 & -3\end{array}\right),\left(\begin{array}{ll}0 & 1 \\ 1 & 1\end{array}\right)\)
d) \(\left(\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right),\left(\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right),\left(\begin{array}{cc}-2 & -2 \\ -2 & 1\end{array}\right)\)
e) \(\left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right),\left(\begin{array}{ll}-1 & -1 \\ -1 & -1\end{array}\right),\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\)
f) \(\left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right),\left(\begin{array}{cc}-1 & 2 \\ 2 & -1\end{array}\right),\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\)