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Ordered pairs of scalars are sometimes called 2-dimensional vectors. (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space.) The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples (ordered lists of n objects).
Let V and W be vector spaces over a field (or more generally, modules over a ring) and let T be a linear map from V to W.If 0 W is the zero vector of W, then the kernel of T is the preimage of the zero subspace {0 W}; that is, the subset of V consisting of all those elements of V that are mapped by T to the element 0 W.
Vector-based devices, such as the vector CRT and the pen plotter, directly control a drawing mechanism to produce geometric shapes. Since vector display devices can define a line by dealing with just two points (that is, the coordinates of each end of the line), the device can reduce the total amount of data it must deal with by organizing the ...
[13] [14] [12] A simpler example of a bound vector is the translation vector from an initial point to an end point; in this case, the bound vector is an ordered pair of points in the same position space, with all coordinates having the same quantity dimension and unit (length an meters).
The notion of greatest common divisor can more generally be defined for elements of an arbitrary commutative ring, although in general there need not exist one for every pair of elements. [ 26 ] If R is a commutative ring, and a and b are in R , then an element d of R is called a common divisor of a and b if it divides both a and b (that is, if ...
A vector pointing from A to B. In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector [1] or spatial vector [2]) is a geometric object that has magnitude (or length) and direction. Euclidean vectors can be added and scaled to form a vector space.
In mathematics and physics, vector notation is a commonly used notation for representing vectors, [1] [2] which may be Euclidean vectors, or more generally, members of a vector space. For denoting a vector, the common typographic convention is lower case, upright boldface type, as in v .
A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space.