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The ring of integers of a number field K, denoted O_K, is the set of algebraic integers in K, which is a ring of dimension d over Z, where d is the extension degree of K over ...

A bounded linear operator T in B(H) on a Hilbert space H is said to be cyclic if there exists some vector v in H for which the set of orbits ...

The vector triple product identity Ax(BxC)=B(A·C)-C(A·B). This identity can be generalized to n dimensions,

A change of coordinates matrix, also called a transition matrix, specifies the transformation from one vector basis to another under a change of basis. For example, if ...

The tail of a vector AB^-> is the initial point A, i.e., the point at which the vector originates. The tails of a statistical distribution with probability density function ...

Also known as the Serret-Frenet formulas, these vector differential equations relate inherent properties of a parametrized curve. In matrix form, they can be written [T^.; ...

Let V=R^k be a k-dimensional vector space over R, let S subset V, and let W={w in V:w·n^^=0} be a subspace of V of dimension k-1, where n^^ is a unit normal vector of W. Then ...

Let K be a field of arbitrary characteristic. Let v:K->R union {infty} be defined by the following properties: 1. v(x)=infty<=>x=0, 2. v(xy)=v(x)+v(y) forall x,y in K, and 3. ...

The name for the set of integers modulo m, denoted Z/mZ. If m is a prime p, then the modulus is a finite field F_p=Z/pZ.

Let (K,|·|) be a valuated field. The valuation group G is defined to be the set G={|x|:x in K,x!=0}, with the group operation being multiplication. It is a subgroup of the ...