Sizes of fixed point spaces for elements in linear groups have a long history. These types of results include the classification of psuedoreflection groups and Frobenius complements. I will talk about recent joint works with Maroti and Malle which answer two conjectures from Peter Neumann's 1966 thesis. The first conjecture has to do with the average dimension of a fixed space of an element in a finite (irreducible) linear group. The second has to do with the minimal dimension of a fixed space of some element in an irreducible (possibly infinite) linear group. I will also discuss some related results that come out of the techniques of the proof.