311 – Extra credit exercise

This optional exercise is due Thursday, February 12, at the beginning of lecture.

Show that the projective plane obtained through the vector space construction applied to the field of two elements is (isomorphic to) the Fano plane.

I review the construction: To form we start with a field and consider .

A point in the projective plane is just a line in through the origin. Note that any such line is obtained by fixing a non-zero element of and considering all its scalar multiples, all the vectors with .

(When , the only values of are and , so a line consists of only two points, one of which is the origin.)

A line in the projective plane is actually the collection of lines contained in a plane in through the origin. In other words, a set of points (that is, of lines through the origin) is a line iff there is a plane in that contains the lines that are the elements of and no others. Note that any such plane can be described by fixing coefficients in , not all equal to zero, and considering the set of vectors such that .

To show that when  this construction produces the Fano plane it is not enough to indicate that the resulting space consists of points and lines, you also need to verify that these can be identified with the points and lines of the Fano plane in such a way that all incidence relations in the Fano plane are verified in this space and no, others.

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