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More precisely, a vector bundle over X is a topological space E equipped with a continuous map : such that for every x in X, the fiber π −1 (x) is a vector space. The case dim V = 1 is called a line bundle. For any vector space V, the projection X × V → X makes the product X × V into a "trivial" vector bundle.
Both vector addition and scalar multiplication are trivial. A basis for this vector space is the empty set, so that {0} is the 0-dimensional vector space over F. Every vector space over F contains a subspace isomorphic to this one. The zero vector space is conceptually different from the null space of a linear operator L, which is the kernel of L.
Free objects are the direct generalization to categories of the notion of basis in a vector space. A linear function u : E 1 → E 2 between vector spaces is entirely determined by its values on a basis of the vector space E 1. The following definition translates this to any category.
The components of a vector are often represented arranged in a column. By contrast, a covector has components that transform like the reference axes. It lives in the dual vector space, and represents a linear map from vectors to scalars. The dot product operator involving vectors is a good example of a covector.
If V is a vector space over a field K, a subset W of V is a linear subspace of V if it is a vector space over K for the operations of V.Equivalently, a linear subspace of V is a nonempty subset W such that, whenever w 1, w 2 are elements of W and α, β are elements of K, it follows that αw 1 + βw 2 is in W.
One solution using the even-odd rule is to transform (complex) polygons into simpler ones that are even-odd-equivalent before the intersection check. [10] This, however, is computationally expensive. It is less expensive to use the fast non-zero winding number algorithm, which gives the correct result even when the polygon overlaps itself.
A vector space with an orientation selected is called an oriented vector space, while one not having an orientation selected, is called unoriented. In mathematics , orientability is a broader notion that, in two dimensions, allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left ...
Codimension is a relative concept: it is only defined for one object inside another. There is no “codimension of a vector space (in isolation)”, only the codimension of a vector subspace. If W is a linear subspace of a finite-dimensional vector space V, then the codimension of W in V is the difference between the dimensions: [1]