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The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
If V is a vector space over a field K, a subset W of V is a linear subspace of V if it is a vector space over K for the operations of V.Equivalently, a linear subspace of V is a nonempty subset W such that, whenever w 1, w 2 are elements of W and α, β are elements of K, it follows that αw 1 + βw 2 is in W.
The same vector can be represented in two different bases (purple and red arrows). In mathematics, a set B of vectors in a vector space V is called a basis (pl.: bases) if every element of V may be written in a unique way as a finite linear combination of elements of B.
Let the field K be the set R of real numbers, and let the vector space V be the Euclidean space R 3. Consider the vectors e 1 = (1,0,0), e 2 = (0,1,0) and e 3 = (0,0,1). Then any vector in R 3 is a linear combination of e 1, e 2, and e 3. To see that this is so, take an arbitrary vector (a 1,a 2,a 3) in R 3, and write:
The above defined vector v ∗ ∈ V ∗ is said to be the dual vector of . In an infinite dimensional Hilbert space , analogous results hold by the Riesz representation theorem . There is a mapping V ↦ V ∗ from V into its continuous dual space V ∗ .
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. Coordinates are always ...
A four-vector A is a vector with a "timelike" component and three "spacelike" components, and can be written in various equivalent notations: [3] = (,,,) = + + + = + = where A α is the magnitude component and E α is the basis vector component; note that both are necessary to make a vector, and that when A α is seen alone, it refers strictly to the components of the vector.
Characters are class functions, that is, they each take a constant value on a given conjugacy class. More precisely, the set of irreducible characters of a given group G into a field F form a basis of the F-vector space of all class functions G → F. Isomorphic representations have the same characters.