ring and module problem

Question

Let $$F=\mathbb{R}$$ $$V=\mathbb{R}^{4}$$ consider two matrices $$S1=\begin{vmatrix} 0&-1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& -1\\ 0&0 & 1 & 0 \end{vmatrix} , S2=\begin{vmatrix} 0&-0& -1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ 0&-1 & 0 & 0 \end{vmatrix}$$ and put $S$={$S1$,$S2$} $$ \forall(\neq 0) v\in V $$ , $$ S,V \notin Fv$$ show that {$V,S1V,S2V,S1S2V$} is linearly independent over $F$ in $V$. I can't understand solved by division ring. I need some your help. please

Answer 1

Take the four matrices $I=\begin{vmatrix} 1&0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0&0 & 0 & 1 \end{vmatrix},S_1=\begin{vmatrix} 0&-1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& -1\\ 0&0 & 1 & 0 \end{vmatrix}$ and $ \\S_2=\begin{vmatrix} 0& 0& -1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ 0&-1 & 0 & 0 \end{vmatrix},S_1S_2=\begin{vmatrix} 0&0& 0& -1\\ 0& 0& -1& 0\\ 0& 1& 0& 0\\ 1 & 0 & 0 & 0 \end{vmatrix}$.

It's easy to verify that $S_1^2=S_2^2=(S_1S_2)^2=S_1S_2(S_1S_2)=-I$. This suggests we should map $S_1\mapsto 1$, $S_2\mapsto i$, $S_1S_2\mapsto k$ from the $\Bbb R$-span of these matrices into the quaternions $\Bbb H$, and verify that this mapping is an isomorphism. This way, we've shown that the ring formed by the span of these matrices is just a copy of Hamilton's quaternions $\Bbb H$, which is a division ring.

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Notes:

"Show that $\{V,S1V,S2V,S1S2V\}$ is linearly independent over $F$ in $V$": I guess you have just not carefully written what you mean. A set of vectors can be linearly independent or not, but you haven't written a set of vectors, you've written a set of four sets of vectors. It would make sense to say "show $\{I,S_1,S_2,S_1S_2\}$ is a linearly independent set".

The following part is also very confused:

put $S$={$S1$,$S2$} $ \forall(\neq 0) v\in V $ ,$ S,V \notin Fv$

Perhaps if you elaborate more in your comments or your post, I can offer more help on that part too.