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Math Help - Cosets form a partition of X

  1. #1
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    Cosets form a partition of X

    Hello guys, I hope I'm posting in the right place.

    I'm having problems solve the following problem: Let Y be a subspace of a vector space X. Show that the distinct cosets x + Y (x in X) form a partition of X.

    I don't quite understand how these cosets work so I couldn't think of any way to approach this problem.
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  2. #2
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    Re: Cosets form a partition of X

    Odd, I thought I was pretty good at linear algebra but I also thought that "cosets" was a topic from group theory! But, of course, the set of vectors, with the single operation of addition is a group so the 'coset' x+ Y, for a given x, is the set of all vectors of the form x+ y where y is any vector in Y. To show that x+ Y is a "partition" of X, we need to show that every vector in one and only one of those sets.

    Let v be any vector in X. Choose any vector y in Y, let x= z- y.

    Now, suppose v is in both x+ Y and x'+ Y, with x\ne x'. That is, v= x+ y_1 and v= x'+ y_2 with both y_1 and y_2 in Y. So we have x+ y_1= x'+ y_2 from which it follows that x- x'= y_2- y_1. What does that tell you?
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