basicsettheory

basicsettheory - SOME REMARKS ABOUT SET NOTATION MA265 1...

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SOME REMARKS ABOUT SET NOTATION MA265, SECTIONS 41, 52 1. Initial comments In linear algebra, we have to deal very often with sets of objects in a more complex way than you have probably seen before. In case it helps, I’m writing some notes to help ease the transition. This is something you will need to know at the very least to understand homework, test, and exam questions about vector spaces, subspaces, and bases, but more generally you will need to understand this in order to understand mathematics properly. Set theory isn’t quite exactly the same thing as mathematics, but it is an extremely fundamental part of mathematics. 2. Set notation and examples using subspaces of R 3 When I write something like R 3 = a b c , where a, b, c R what I mean is that R 3 is (1) A collection of things (2) The things in R 3 are matrices with 3 components, labeled a, b, c (3) The letters a, b, c refer to real numbers, and those real numbers have no restrictions imposed on them. What this means is that if someone hands you such a thing, with real numbers in the place of a, b, c , it’s in R 3 . So all of these things (and many many more) would be in R 3 : 1 - 2 0 , 0 0 0 , - 2 1 1 , 1 1 1 , π e = 2 . 718281828 ... - 1 , 2 3 5 R 3 are in R 3 , but things like 1 - 2 0 , 1 - 2 , x y z 0 are not ( / R 3 ). Sometimes, certain statements are obvious from context, like in real linear algbebra we would know that a, b, c represent real numbers, so we would simply write R 3 = a b c
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