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Unformatted text preview: NOTES ON SECTION 4 . 2 MA265, SECTIONS 41, 52 Some key words (vector space) axiom (Real) vector space vectors scalars addition scalar multiplication zero vector negative of a vector polynomial function space C ( R , R ) M n P [ x ] P n [ x ] 1. Vector spaces Suppose V is a set (a collection of things), together with two “operations” ( ⊕ , ), by which I mean: (1) Given two things v 1 ,v 2 in V , there is anothing thing called v 1 ⊕ v 2 in V as well. (2) Given something v in V , and a real number (scalar) r , there is another thing r v in V as well. Remark 1. In practice, ⊕ and are things we understand and usually denote by the symbols + , · instead. We use these symbols to clarify that these are abstractions of the usual notion of + , · . It might be worth pointing out that the symbols + , · that we are used to actually have multiple meanings that we allow ourselves to confuse. To illustrate what I mean, I point out that adding two integers is really a different thing than adding two fractions (indeed, it seems that learning to add fractions causes some people much frustration, which is often forgotten soon after), but we call them both +. Of course, this is justified because the integers “embed” in the rational numbers, but the point that it is a new + shouldn’t be forgotten. Same comment applies when we think about real numbers. 1 Definition 2. Take ( V, ⊕ , ) as described above. We call this data a vector space if the following axioms are satisfied (where u,v,w are arbitrary elements of V and r,s are arbitrary scalars): V1: u ⊕ ( v ⊕ w ) = ( u ⊕ v ) ⊕ w . V2: There is a special vector such that for every v ∈ V we have + v = v + = v . V3: For any vector u , there is another vector- u such that u ⊕ (- u ) = (- u ) ⊕ u = . V4: v ⊕ w = w ⊕ v V5: r ( u ⊕ v ) = r u ⊕ r v V6: ( r + s ) v = r v ⊕ s v V7: r ( s v ) = ( rs ) v V8: 1 v = v Remark 3. Implicit in the above is the following convention: whenever the order of operations is ambiguous, perform the operation before the ⊕ operation....
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