I just received a message from the Linear Algebra lecturer
that there are a couple of typos on Linear Algebra Sheet 1.
In Question 4, you can assume that V is finite-dimensional.
Question 5 should read as follows:
Question 5. Let V be a vector space over a field F and let E_1
and E_2 be projections on V.
(i) Show that E_1 + E_2 is a projection if and only
if E_1E_2 + E_2E_1 = 0.
(ii) Suppose that Char F noteq 2. Prove that E_1 + E_2 is a
projection if and only if E_1E_2 = E_2E_1 = 0. [Hint: consider
E_1E_2E_1 in two different ways.] Assuming this is the case, find the
kernel and image of E_1 + E_2 in terms of those of E_1 and E_2.
(iii) What can be said if Char F = 2?