Quotient - The Quotient vector Space Suppose V is a vector...

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Unformatted text preview: The Quotient vector Space Suppose V is a vector space over K and U V is a subspace. We will describe a construction of the quotient vector space V/U . But first we will discuss equivalence relations. If S is a set then a relation on S is some way of relating elements of S . The expression x y means x is related to y . For example if S is all the people in the world, then x y might mean x = y or x is a brother or sister of y and x y might mean x is a mother of y . Both and are relations. We say a relation is an equivalence relation if a) x x for all x S . b) If x y then y x . c) If x y and y z then x z . Thus in the examples above is an equivalence relation but is not. For a mathematical example, let S be the integers and say that x y if and only if x- y is even. This is an equivalence relation. If you have an equivalence relation then S can be divided up into equivalence classes. An equivalence class is a set of the form [...
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