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In page Real projective space:

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The projective n {\displaystyle n} -space is in fact diffeomorphic to the submanifold of R ( n + 1 ) 2 {\displaystyle \mathbb {R} ^{(n+1)^{2}}} consisting of all symmetric ( n + 1 ) × ( n + 1 ) {\displaystyle (n+1)\times (n+1)} matrices of trace 1 that are also idempotent linear transformations.[citation needed]