# Can Three Non-Collinear Points Always Define a Projective Plane?

• Euler2718
In summary, the problem states that given a projective space P(W) with dimension greater than or equal to 2 and three non-colinear projective points, it is necessary to prove the existence of a projective plane containing all three points. By defining a three-dimensional vector space V spanned by the three points, it can be shown that P(V) is a subspace of P(W) and thus a plane containing the three points.
Euler2718

## Homework Statement

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

## The Attempt at a Solution

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For $n=2$ my reasoning was: WLOG assume $[v_{1}],[v_{2}],[v_{3}]$ are distinct. Then, define $s = \{ v_{1},v_{2},v_{3} \}$ which spans a vector space, S. The dimension here is 3, and since $v_{1},v_{2},v_{3}\in S$ it follows that $[v_{1}],[v_{2}],[v_{3}]\in P(S)$, 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 $P(W)$ had dimension n (>2) , $s = \{ v_{1},v_{2},v_{3} \}$ 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?

First, there is no need to ask that the points are distinct. They are non colinear, so projectively independent. It also follows that ##v_1, v_2, v_3## must be linearly independent in ##W##, so they span a three-dimensional vectorspace. Now, it is sufficient to define ##V = span\{v_1,v_2,v_3\}## and then say that ##P(V)## is the subspace of ##P(W)## you are looking for. Note that ##P(V)## is a plane, as it has dimension 2.

Euler2718

## What is a projective plane proof?

A projective plane proof is a type of mathematical proof that involves using the properties of a projective plane to prove a statement or theorem. The projective plane is a geometric concept that extends the properties of a traditional plane to include points at infinity.

## What are the properties of a projective plane?

A projective plane has several key properties, including: every two distinct lines intersect at exactly one point, every two distinct points lie on exactly one line, and there are no parallel lines.

## How is a projective plane proof different from a traditional proof?

A projective plane proof differs from a traditional proof in that it uses the unique properties of a projective plane, such as the absence of parallel lines, to prove a statement. It also extends the traditional concepts of points, lines, and planes to include points at infinity.

## What types of problems can be solved using projective plane proofs?

Projective plane proofs can be used to solve a variety of problems in geometry, including proving theorems about lines and points, determining the intersections of lines and planes, and solving geometric constructions.

## Are there any limitations to using projective plane proofs?

While projective plane proofs can be a powerful tool in solving geometric problems, they do have some limitations. For example, projective plane proofs can only be used for problems that involve points, lines, and planes, and may not be applicable to other mathematical concepts.

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