 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/digital-systems-linear-algebra-introduction-to-formal-proof/5/e/i/solution

Prove both inclusions. If

v

belongs to the column space of

I - P

, then

v = (I - P) u

for some

u

and

P v = (P - P^{2}) u = 0

. Thus

v \in N (P)

Conversely, if

P v = 0

, then

v = (I - P) v

, which explicitly writes

v

I - P

. Hence

v \in C (I - P)

. The two inclusions prove $N (P) = C (I - P)$ . No orthogonality assumption was used.

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