Talk:4.3.1/@comment-5149782-20120629150152/@comment-5197940-20120629210256

Made a typo in $$M_2 $$, which is now corrected.

If you're asking why they form a basis, it's easy to check they span and are linearly independent. If you're asking where they come from, I think the best way to see that is to decompose the symmetric matrix $$\begin{bmatrix}

x & y \\

y & z \\

\end{bmatrix}

$$ into the linear combination $$xM_1 + yM_2 + zM_3 $$.