These are solved by considering as an Euclidean vector space.
2.
If the underlying symmetric form, for example, the scalar product on Euclidean vector spaces.
3.
Non-linear dimensionality reduction algorithms attempt to map high-dimensional data onto a low-dimensional Euclidean vector space.
4.
In a geometric algebra generated by a Euclidean vector space of dimension 2 or 3, all sums of 2-blades may be written as a single 2-blade.
5.
Where we omitted the t parameter, and regard \ mathbf { v } and \ mathbf { r } as elements of the same 3-dimensional Euclidean vector space V.
6.
Therefore, for a hermitian vector space the vector space ? ( as well as its complex conjugate ? " V " ) is a spinor space for the underlying real euclidean vector space.
7.
A geometric algebra generated by a 4-dimensional Euclidean vector space illustrates the point with an example : The sum of any two blades with one taken from the XY-plane and the other taken from the ZW-plane will form a 2-vector that is not a 2-blade.
8.
As with the term " Euclidean space ", " pseudo-Euclidean space " may refer to either an affine space or a vector space, though the latter may also be referred to as a "'pseudo-Euclidean vector space "'( see point vector distinction ).
9.
While seldom used explicitly, the geometric view of the complex numbers is implicitly based on its structure of a Euclidean vector space of dimension 2, where the inner product of complex numbers and is given by \ Re ( w \ overline { z } ); then for a complex number its absolute value | | coincides with its Euclidean norm, and its argument with the angle turning from 1 to.
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