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Matrices, Arrays and Determinants

  • C. C. Mac Duffee
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Part of the Ergebnisse der Mathematik und Ihrer Grenƶgebiete book series (MATHE1, volume 5)

Abstract

Linear algebra. A linear algebra e of order n over a field p is composed of two or more numbers α, β, γ, and three operations, addition (+), multiplication (ϗ) and scalar multiplication such that α β, α × β, α a, α are uniquely defined numbers of e, where a is in p. It is further assumed that addition is commutative and associative, and that multiplication is distributive with respect to addition. If a and b are in p it is assumed that
$$a\alpha {\text{} } = {\text{} }\alpha a,{\text{} }a(b\alpha){\text{} } = {\text{} }(ab)\alpha,{\text{} }(a\alpha)(b\beta){\text{} } = {\text{} }(ab){\text{} }(a\beta)$$
$$(a{\text{} } + {\text{} }b)\alpha {\text{} } = {\text{} }a\alpha {\text{} } + {\text{} }b\alpha,{\text{} }a(\alpha {\text{} } + {\text{} }\beta){\text{} } = {\text{} }a\alpha {\text{} } + {\text{} }a\beta$$
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Notes

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