Kunze Linear Algebra · Problem 6.6.10

Elementary Canonical Forms

Mathematics — Methods · projections-sum-identity-orthogonality

Problem

Let

F

be a subfield of the complex numbers (or, a field of characteristic zero). Let

V

be a finite-dimensional vector space over

F

. Suppose that

E_1, \ldots, E_k

are projections of

V

and that

E_1 + \cdots + E_k = I

. Prove that

E_i E_j = 0

for

i \neq j

. (Hint: Use the trace function and ask yourself what the trace of a projection is.)