By Shun-Ichi Amari, Hiroshi Nagaoka

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**Sample text**

A vector space E = {0} is called a nontrivial vector space. ). The dimension of the vector space E is denoted by dim(E). The direct sum of two vector spaces U,V is denoted by U ⊕V . The dual of a vector space E is denoted by E ∗ . The kernel of a linear map f : E → F is denoted by Ker f , and the image by Im f . The transpose of a matrix A is denoted by A . The identity function is denoted by id, and the n × n-identity matrix is denoted by In , or I. The determinant of a matrix A is denoted by det(A) or D(A).

Riemann studied spherical spaces of higher dimension, and showed that their geometry is non-Euclidean. Finally, Cayley (1821–1895) and especially Klein (1849–1925) reached a clear understanding of the various geometries and their relationships. Basically, all geometries can be viewed as embedded in a universal geometry, projective geometry. Projective geometry itself is non-Euclidean, since two coplanar lines always intersect in a single point. Projective geometry was developed in the nineteenth century, mostly by Monge, Poncelet, Chasles, Steiner, and Von Staudt (but anticipated by Kepler (1571–1630) and Desargues (1593–1662)).

1 that it is possible to make sense of linear combinations of points, and even mixed linear combinations of points and vectors. − → − → Any vector space E has an afﬁne space structure speciﬁed by choosing E = E , − → and letting + be addition in the vector space E . We will refer to the afﬁne structure − →− → − → E , E , + on a vector space E as the canonical (or natural) afﬁne structure on − → E . In particular, the vector space Rn can be viewed as the afﬁne space Rn , Rn , + , denoted by An .