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The dimension of an affine space is defined as the dimension of the vector space of its translations. An affine space of dimension one is an affine line. An affine space of dimension 2 is an affine plane. An affine subspace of dimension n – 1 in an affine space or a vector space of dimension n is an affine hyperplane.
The multivariate normal distribution is said to be "non-degenerate" when the symmetric covariance matrix is positive definite. In this case the distribution has density [ 5 ] where is a real k -dimensional column vector and is the determinant of , also known as the generalized variance.
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a ...
The 1-dimensional Berkovich affine space is called the Berkovich affine line. When k {\displaystyle k} is an algebraically closed non-Archimedean field, complete with respects to its valuation, one can describe all the points of the affine line.
For the group of affine transformations on the parameter space of the normal distribution, the right Haar measure is the Jeffreys prior measure. [13] Unfortunately, even right Haar measures sometimes result in useless priors, which cannot be recommended for practical use, like other methods of constructing prior measures that avoid subjective ...
Hilbert space. The state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space. In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear ...
The geometric notions of an affine space, projective space, convex set, and cone have related notions of basis. [4] An affine basis for an n-dimensional affine space is + points in general linear position. A projective basis is + points in general position, in a projective space of dimension n.