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They are elements in R 2 and R 4; ... is a formal way of adding products to any vector space to obtain an algebra. [88] As a vector space, it is ...
Every algebra over a field is a vector space, but elements of an algebra are generally not called vectors. However, in some cases, they are called vectors, mainly due to historical reasons. Vector quaternion, a quaternion with a zero real part; Multivector or p-vector, an element of the exterior algebra of a vector space.
In standard matrix notation, each element of R n is typically written as a column vector = [] and sometimes as a row vector: = []. The coordinate space R n may then be interpreted as the space of all n × 1 column vectors , or all 1 × n row vectors with the ordinary matrix operations of addition and scalar multiplication .
In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. For a vector , v → {\displaystyle {\vec {v}}\!} , adding two matrices would have the geometric effect of applying each matrix transformation separately onto v → {\displaystyle {\vec {v}}\!} , then adding the transformed vectors.
More specifically, they can be characterized as orthogonal matrices with determinant 1; that is, a square matrix R is a rotation matrix if and only if R T = R −1 and det R = 1. The set of all orthogonal matrices of size n with determinant +1 is a representation of a group known as the special orthogonal group SO( n ) , one example of which is ...
Let p = (p 1, p 2) be an element of W, that is, a point in the plane such that p 1 = p 2, and let c be a scalar in R. Then cp = (cp 1, cp 2); since p 1 = p 2, then cp 1 = cp 2, so cp is an element of W. In general, any subset of the real coordinate space R n that is defined by a homogeneous system of linear equations will yield a subspace.
Vectors are defined in spherical coordinates by (r, θ, φ), where r is the length of the vector, θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and; φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π).
The term vector was coined by W. R. Hamilton around 1843, as he revealed quaternions, a system which uses vectors and scalars to span a four-dimensional space. For a quaternion q = a + b i + c j + d k, Hamilton used two projections: S q = a , for the scalar part of q , and V q = b i + c j + d k, the vector part.