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Unformatted text preview: Notes on linear algebra (Monday 17th October. 2016, 23:10) page 188 is concerned the requirements on it are literally the same in the two definitions104 (commutativity of addition associativity of addition neutrality of 0 and existence of additive inverses). However the similarity ends here: The binary operation - (”multiplication") in Definition 2.50 works differently from the binary operation - (”scaling”) in Definition 4.2. The former takes two inputs in the commutative ring, whereas the latter takes one input in IR and one input in the vector space. The axioms still have certain similarities, but they should not fool you into believing that commutative rings are vector spaces (or vice versa). If you compare our Definition 4.2 with other definitions of a ”vector space” you find in the literature (for example, [LaNaSc16, Definition 4.1.1], [Oleha06, Defini- tion 2.1] or [Heffer16, Definition Two.I.1]), you will notice that they are slightly different: For example, [LaNaSc16, Definition 4.1.1], [Oleha06, Definition 2.1] or [Heffer16, Definition Two.I.1] are lacking our properties (i) and (j), whereas [Kowals16, Definition 2.3.1] is missing our properties (d) and (j). However, the def- initions are nevertheless equivalent (i.e., they define precisely the same notion of a vector space). The reason for that is some of the properties we required in Defini- tion 4.2 are redundant (i.e., they follow from the other properties, so that nothing changes if we leave them out). For example: Proposition 4.3. Property (d) in Definition 4.2 follows from properties (e), (h) and (1). Thus, we could leave out property (d) from the definition. ...
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