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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This note was uploaded on 07/19/2008 for the course MATH 110 taught by Professor Gurevitch during the Summer '08 term at Berkeley.

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