- #1

Euler2718

- 90

- 3

## Homework Statement

Let [itex]P(W)[/itex] be a projective space whose dimension is greater than or equal to 2 and let three non-colinear projective points, [itex][v_{1}],[v_{2}],[v_{3}]\in P(W) [/itex]. Prove that there is a projective plane in [itex]P(W)[/itex] containing all three points.

## Homework Equations

## The Attempt at a Solution

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For [itex]n=2[/itex] my reasoning was: WLOG assume [itex][v_{1}],[v_{2}],[v_{3}][/itex] are distinct. Then, define [itex]s = \{ v_{1},v_{2},v_{3} \}[/itex] which spans a vector space, S. The dimension here is 3, and since [itex]v_{1},v_{2},v_{3}\in S[/itex] it follows that [itex][v_{1}],[v_{2}],[v_{3}]\in P(S) [/itex], has dimension two and is thus a plane containing the points.

If this idea is correct, then I am at the point now where I am trying to prove it for all dimensions greater than 2. I was thinking that is [itex]P(W)[/itex] had dimension n (>2) , [itex]s = \{ v_{1},v_{2},v_{3} \}[/itex] still spans a three dimensional vector space which would be a subspace of W. So the projective space of S would be a linear subspace of the projective space of W?