# Affine plane

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

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

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

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