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A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector.
Scalar multiplication of a vector by a factor of 3 stretches the vector out. The scalar multiplications −a and 2a of a vector a. In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra [1] [2] [3] (or more generally, a module in abstract algebra [4] [5]).
In the geometrical setting, a non-zero element of the top exterior power () (which is a one-dimensional vector space) is sometimes called a volume form (or orientation form, although this term may sometimes lead to ambiguity). The name orientation form comes from the fact that a choice of preferred top element determines an orientation of the ...
Scalar division is performed by multiplying the vector operand with the multiplicative inverse of the scalar operand. This can be represented by the use of the fraction bar or division signs as operators. The quotient of a vector v and a scalar c can be represented in any of the following forms:
Vector addition is just matrix addition and scalar multiplication is defined in the obvious way (by multiplying each entry by the same scalar). The zero vector is just the zero matrix. The dimension of F m×n is mn. One possible choice of basis is the matrices with a single entry equal to 1 and all other entries 0.
They are functions that reflect the vector space structure, that is, they preserve sums and scalar multiplication: (+) = + (), = for all and in , all in . [37] An isomorphism is a linear map f : V → W such that there exists an inverse map g : W → V , which is a map such that the two possible compositions f ∘ g : W → W and g ∘ f : V ...
The product zz* is a quadratic form for each of the complex numbers, split-complex numbers, and dual numbers. For z = x + ε y, the dual number form is x 2 which is a degenerate quadratic form. The split-complex case is an isotropic form, and the complex case is a definite form.
Since the quadratic form is a scalar quantity, = (). Next, by the cyclic property of the trace operator, [ ()] = [ ()]. Since the trace operator is a linear combination of the components of the matrix, it therefore follows from the linearity of the expectation operator that