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The transpose (indicated by T) of any row vector is a column vector, and the transpose of any column vector is a row vector: […] = [] and [] = […]. The set of all row vectors with n entries in a given field (such as the real numbers ) forms an n -dimensional vector space ; similarly, the set of all column vectors with m entries forms an m ...
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.
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space. [1] A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane.
The column space of this matrix is the vector space spanned by the column vectors. In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation.
The components of this vector transform between themselves contravariantly under rotations in space. Similarly, a dual (or co-) vector field attaches a dual vector to each point of space, and the components of each dual vector transform covariantly. tensor fields, (such as the stress tensor of a crystal) specified by a tensor at each point of ...
In the simple case where we consider the vector space , a ket can be identified with a column vector, and a bra as a row vector. If, moreover, we use the standard Hermitian inner product on C n {\displaystyle \mathbb {C} ^{n}} , the bra corresponding to a ket, in particular a bra m | and a ket | m with the same label are conjugate transpose .
A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier". [4] It was first used by 18th century astronomers investigating planetary revolution around the Sun. [5] The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B.
A particular choice of the scalar and vector potentials is a gauge (more precisely, gauge potential) and a scalar function ψ used to change the gauge is called a gauge function. [ citation needed ] The existence of arbitrary numbers of gauge functions ψ ( r , t ) corresponds to the U(1) gauge freedom of this theory.