Linear Space

S. N. Afriat
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S. N. Afriat: University of Siena

Chapter 2 in Linear Dependence, 2000, pp 23-38 from Springer

Abstract: Abstract A commutative group L, with group operation + and identity o,is considered together with a field K, and an operation between them by which any $$t \in K,x \in L$$ Determine $$xt \in L$$ ,with the properties $$\begin{array}{*{20}{c}} {xI = x,\left( {x + y} \right)t = xt + yt,} \\ {x\left( {s + t} \right) = xs + xt,x\left( {st} \right) = \left( {xs} \right)t.} \\ \end{array}$$ The system formed by K and L and the operation between them with these properties defines a linear space L over K. Elements of L are vectors, and those of K are the scalars, and the operations are called vector addition and scalar multiplication.

Keywords: Linear Space; Linear Subspace; Scalar Multiplication; Factor Space; Affine Space (search for similar items in EconPapers)
Date: 2000
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DOI: 10.1007/978-1-4615-4273-5_3

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