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Then, the coordinates of a vector form a sequence similarly indexed, and a vector is completely characterized by the sequence of coordinates. An ordered basis, especially when used in conjunction with an origin, is also called a coordinate frame or simply a frame (for example, a Cartesian frame or an affine frame).
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. [1] An easy example may be a position such as (5, 2, 1) in a 3-dimensional Cartesian coordinate system with the basis as the axes of this system.
If two different bases are considered, the coordinate vector that represents a vector v on one basis is, in general, different from the coordinate vector that represents v on the other basis. A change of basis consists of converting every assertion expressed in terms of coordinates relative to one basis into an assertion expressed in terms of ...
tangent basis e 1, e 2, e 3 to the coordinate curves (left), dual basis, covector basis, or reciprocal basis e 1, e 2, e 3 to coordinate surfaces (right), in 3-d general curvilinear coordinates (q 1, q 2, q 3), a tuple of numbers to define a point in a position space. Note the basis and cobasis coincide only when the basis is orthonormal. [1 ...
where δs is the displacement vector between the point P and a nearby point Q whose coordinate separation from P is δx α along the coordinate curve x α (i.e. the curve on the manifold through P for which the local coordinate x α varies and all other coordinates are constant). [1]
One also says that the n-tuple of the coordinates is the coordinate vector of v on the basis, since the set of the n-tuples of elements of F is a vector space for componentwise addition and scalar multiplication, whose dimension is n.
We now face three different basis sets commonly used to describe vectors in orthogonal coordinates: the covariant basis e i, the contravariant basis e i, and the normalized basis ê i. While a vector is an objective quantity, meaning its identity is independent of any coordinate system, the components of a vector depend on what basis the vector ...
Every vector a in three dimensions is a linear combination of the standard basis vectors i, j and k.. In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as or ) is the set of vectors, each of whose components are all zero, except one that equals 1. [1]