quiz x solutions

Fact 1 contains under assumption 1 there exists a

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Unformatted text preview: , πi2 , · · · , πij ⊆ π1 , π2 , · · · , πk consisting of j distinct points is co- j − 2 -planar, namely, no j − 2 -plane contains all those points πi1 , πi2 , · · · , πij . Fact. (1) contains Under Assumption 1, there exists a unique k − 1 -plane Λ that π1 , π2 , · · · , πk as its subset. We call Λ the linear span of the set π1 , π2 , · · · , πk . (2) Consider the dual projective n-space Pn ∨ to the original Pn . Let Π1 , Π2 , · · · , Πk be the codimension 1 hyperplanes correspond to π1 , π2 , · · · , πk , under duality, respectively. be the intersection of Π1 , Π2 , · · · , Πk . Then Λ ⊆ Pn correspond to each other under duality. (3) projective n-space inside Pn ∨ which Let Λ′ ⊆ Pn ∨ and Λ′ ⊆ Pn ∨ There exists a codimension 1 hyperplane Π inside Pn that contains π1 , π2 , · · · , πk as its subset. Λ is the intersection of all codimension 1 hyperplanes inside Pn that contains the above set as its subset. 6 (4a) Let aj 0 : aj 1 : aj 2 : aj 3 : · · · : ajn be the coordinate reading of the point πj for j = 1, 2, · · · k . Using those, form the matrix a10 a11 a12 a13 · · · a1n a20 a21 a22 a23 · · · a2n . . . . . .. . . . . . . . . . ak0 ak1 ak2 ak3 · · · akn Then the rank of this matrix equals k . (4b) Conversely, suppose a matrix a10 a20 . . . ak 0 a11 a21 . . . a12 a22 . . . a13 a23 .. . ··· ··· . . . ak 1 ak 2 ak 3 ··· a1n a2n akn of size k × n + 1 , has rank k . In particular, no rows consist entirely of 0s. Form the ratio out of each row, and regard them as k number of points inside Pn . Then those k number of points satisfy Assumption 1. 7...
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