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The linear maps (or linear functions) of vector spaces, viewed as geometric maps, map lines to lines; that is, they map collinear point sets to collinear point sets and so, are collineations. In projective geometry these linear mappings are called homographies and are just one type of collineation.
Formally, let V be a vector space over a field K and W a vector space over a field L. Consider the projective spaces PG(V) and PG(W), consisting of the vector lines of V and W. Call D(V) and D(W) the set of subspaces of V and W respectively. A collineation from PG(V) to PG(W) is a map α : D(V) → D(W), such that: α is a bijection.
a, b, and c are related to the slope of the line, such that the direction vector (a, b, c) is parallel to the line. Parametric equations for lines in higher dimensions are similar in that they are based on the specification of one point on the line and a direction vector.
Moreover, if the entire vector space V can be spanned by the eigenvectors of T, or equivalently if the direct sum of the eigenspaces associated with all the eigenvalues of T is the entire vector space V, then a basis of V called an eigenbasis can be formed from linearly independent eigenvectors of T.
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 ...
The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example).
Let V be the three-dimensional vector space defined over the field F. The projective plane P(V) = PG(2, F) consists of the one-dimensional vector subspaces of V, called points, and the two-dimensional vector subspaces of V, called lines. Incidence of a point and a line is given by containment of the one-dimensional subspace in the two ...
For instance, the Sylvester–Gallai theorem, stating that any non-collinear set of points in the plane has an ordinary line containing exactly two points, transforms under projective duality to the statement that any projective arrangement of finitely many lines with more than one vertex has an ordinary point, a vertex where only two lines cross.