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## Main Question or Discussion Point

In an affine plane of order

*n*, prove that each line contain exactly*n*points- Thread starter yusukered07
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In an affine plane of order *n*, prove that each line contain exactly *n* points

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

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

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

provided that the coefficients [tex] a_{jh}[/tex] satisfy the orthogonality condition

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

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