Now, I've been learning about character theory and I've been building up to showing that the character table is square; the number of irreducible characters is equal to the number of conjugacy classes.
During the proof of this, it was stated that the number of irreducible $G$-spaces is less than or equal to the number of conjugacy classes, with the following reasoning;
In a vector space of dimension $n$, you can't have more than $n$ non-zero vectors which are pairwise orthogonal.
Now, I know this must be referring to the irreducible characters being pairwise orthogonal, and I know the reasoning is clearly true, but I'm not entirely sure how it relates to the proof of the statement?
What vector space are they referring to here? How does it relate to conjugacy classes?