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The following definition is from the book Lectures in Abstract Algebra, by Jacobson. Chapter "Witt's theorem".

Definition Let $\Re$ be a vector space, let $\mathfrak{S}_1$ and $\mathfrak{S}_2$ be its subspaces. Then subspaces $\mathfrak{S}_1$ and $\mathfrak{S}_2$ are $g$-equivalent if there exist a bijective linear map $U: \mathfrak{S}_1 \rightarrow \mathfrak{S}_2$ such that $$g(x_1,y_1) = g(U(x_1),U(y_1))$$ for all $x_1,y_1 \in \mathfrak{S}_1$.

Question: I tried to look for an examples of $g$-equivalent subspaces but I could not find anything, even the definition. Could someone please give an example of a vector space, its $g$-equivalent subspaces and a bijective linear map so that the definition holds.

Thanks in advance

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  • $\begingroup$ For example, if $g$ is a positive definite bilinear form (inner product), then any two $k$ dimensional subspaces are $g$-equivalent. It's not true, however, for semidefinite or indefinite forms. $\endgroup$ – Berci Apr 8 at 15:37
  • $\begingroup$ Thank you for your answer. In this case $U$ can be any bijective linear map? $\endgroup$ – IllariyaS Apr 8 at 16:31
  • $\begingroup$ Any isometric map would do, yes. $\endgroup$ – Berci Apr 8 at 18:35

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