{"title":"Vector Spaces","authors":"","doi":"10.1002/9781119111771.app3","DOIUrl":"https://doi.org/10.1002/9781119111771.app3","url":null,"abstract":"We now begin our treatment of the principal subject matter of this text. We shall see that all of linear algebra is essentially a study of various transformation properties defined on a vector space, and hence it is only natural that we carefully define vector spaces. This chapter therefore presents a fairly rigorous development of (finite-dimensional) vector spaces, and a discussion of their most important fundamental properties. Basically, the general definition of a vector space is simply an axiomatization of the elementary properties of ordinary three-dimensional Euclidean space. A nonempty set V is said to be a vector space over a field F if: (i) there exists an operation called addition that associates to each pair x, y ∞ V a new vector x + y ∞ V called the sum of x and y; (ii) there exists an operation called scalar multiplication that associates to each a ∞ F and x ∞ V a new vector ax ∞ V called the product of a and x; (iii) these operations satisfy the following axioms: (V1) x + y = y + x for all x, y ∞ V. (V2) (x + y) + z = x + (y + z) for all x, y, z ∞ V. (V3) There exists an element 0 ∞ V such that 0 + x = x for all x ∞ V.","PeriodicalId":220953,"journal":{"name":"Semi‐Riemannian Geometry","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124291979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Scalar Product Spaces","authors":"L. H. Loomis, S. Sternberg","doi":"10.1142/9789814583947_0006","DOIUrl":"https://doi.org/10.1142/9789814583947_0006","url":null,"abstract":"","PeriodicalId":220953,"journal":{"name":"Semi‐Riemannian Geometry","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129726599","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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