Proving Lines Contain Exactly n Points in an Affine Plane of Order n

[yusukered07]

In an affine plane of order n, prove that each line contain exactly n points

 

[i_emanuel]

Hi, I will try to help you out :)

Lets consider an n-dimensional Euclidean space E_{n} and by means of abstraction we develop the algebra of general affine tensors.

An orthonormal system e_{j} in E_{n} consists of n mutually orthogonal unit vectors. Any orthonormal system {\bar{e}_{j}} may be obtained from the first by means of the linear transformation

\bar{e}_{j} = \sum^{n}_{h= 1} a_{ j, h} e_{h} (j = 1, ..., n)

provided that the coefficients a_{jh} satisfy the orthogonality condition

\delta _{j,k} = \sum^{n}_{h=1} a_{j,h} a_{k, h} (j, k = 1, ...,n),

...