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A description of the projective geometry can be constructed in the geometric algebra using basic operations. For example, given two distinct points in RP n−1 represented by vectors a and b the line containing them is given by a ∧ b (or b ∧ a). Two lines intersect in a point if A ∧ B = 0 for their bivectors A and B. This point is given ...
Some r-vectors are scalars (r = 0), vectors (r = 1) and bivectors (r = 2). One may generate a finite-dimensional GA by choosing a unit pseudoscalar (I). The set of all vectors that satisfy = is a vector space. The geometric product of the vectors in this vector space then defines the GA, of which I is a member.
When modelling relations between two different classes of objects, bipartite graphs very often arise naturally. For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an affiliation network, a type of bipartite graph used in social network analysis.
In the study of geometric algebras, a k-blade or a simple k-vector is a generalization of the concept of scalars and vectors to include simple bivectors, trivectors, etc. Specifically, a k-blade is a k-vector that can be expressed as the exterior product (informally wedge product) of 1-vectors, and is of grade k. In detail: [1] A 0-blade is a ...
In mathematics, the signature of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and zero eigenvalues of the real symmetric matrix g ab of the metric tensor with respect to a basis.
Plane-based GA usually handles the (3D version of) the middle case here. But we instead choose to have a basis element squaring to 1 or −1 instead of 0, euclidean geometry can be changed to spherical or hyperbolic geometry. In these systems, the points, planes, and lines have the same coordinates that they have in plane-based GA.
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The tangent vectors e 1 and e 2 of a frame on M define smooth functions from E with values in R 3, so each gives a 3-vector of functions and in particular de 1 is a 3-vector of 1-forms on E. The connection form is given by