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 . That is, and with both and in Y. So we have from which it follows that . What does that tell you?