- #36

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Klaas van Aarsen said:What is the matrix of $\sigma_v$ with respect to the basis $(v,w,b_3,b_4)$?

What is the matrix of $\sigma_w$ with respect to the basis $(v,w,b_3,b_4)$?

What is the product of those matrices?

Can we find a more convenient basis so that we get $D$?

It holds that $\sigma_v(v)=-v$ and $\sigma_v(w)=\sigma_v(b_3)=\sigma_v(b_4)=0$, or not?

So we get the matrix $\begin{pmatrix}-1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}$.

Respectively, $\sigma_w(w)=-w$ and $\sigma_w(v)=\sigma_w(b_3)=\sigma_w(b_4)=0$, right?

So we get the matrix $\begin{pmatrix}0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}$.

The product of those matrices is the zero matrix, or not?

:unsure: