# Cosets form a partition of X

• Apr 2nd 2013, 03:45 PM
douglasdc
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.
• Apr 3rd 2013, 02:59 PM
HallsofIvy
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 \$\displaystyle x\ne x'\$. That is, \$\displaystyle v= x+ y_1\$ and \$\displaystyle v= x'+ y_2\$ with both \$\displaystyle y_1\$ and \$\displaystyle y_2\$ in Y. So we have \$\displaystyle x+ y_1= x'+ y_2\$ from which it follows that \$\displaystyle x- x'= y_2- y_1\$. What does that tell you?