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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.
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The space of vectors may be considered a coordinate space where elements are associated with a list of elements from K. The units of the field form a group K × and the scalar-vector multiplication is a group action on the coordinate space by K ×. The zero of the field acts on the coordinate space to collapse it to the zero vector.
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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 ...
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