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Problem: Describe the projective transformations of $FP^n$ that preserve the $FP^{n−1}$ at infinity given by $x_0 = 0$.

My attempt: We have to preserve the points with first coordinate $0$, i.e. the transformation maps $[0, x_1, x_2...x_n]$ to $[0, y_1, ...y_n]$. I can see why we can represent $FP^n$ as the disjoint union of $F^n$ and $FP^{n-1}$ and the latter is basically what we want to preserve. Now I also know that a projective transformation is given by $v$ -> $[Tv]$, where $T:R^n$ -> $R^n$ is injective. I just can't seem to finalize the argument.

Thank you in advance

Rick M

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Hint

Let's work with vector spaces : if $V = F^{n+1}$ with basis $B = (e_0, e_1, \dots, e_n)$ you are asking for matrices $g \in GL(V)$ such that $g(W) = W$ where $W$ has basis $(e_1, \dots, e_n)$. Do you agree ? Do you see how to finish ?

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  • $\begingroup$ Hey, yup this is correct. What I found was that the first row of the matrix of transformation is $(a,0,0,....0)$. Also, that the right-down block matrix can't be 0 and needs to have determinant $\neq 0$. $\endgroup$ – asdf May 1 '17 at 15:00
  • $\begingroup$ Exactly. ${{{}}}$ $\endgroup$ – user171326 May 1 '17 at 15:03
  • $\begingroup$ Wait, is this actually what the question wants us to do? It is a lot simpler than expected. You are the best, really appreciate it! :) $\endgroup$ – asdf May 1 '17 at 15:03
  • $\begingroup$ Sure you're welcome, you did find by yourself :) $\endgroup$ – user171326 May 1 '17 at 15:06

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