 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2019/11/c/iii/solution

Using

M^{2} = M

(I - M)^{2} = I - 2 M + M^{2} = I - M .

Thus

I - M

M (I - M) = (I - M) M = 0

: as linear projections, they select the complementary summands in the decomposition

x = M x + (I - M) x

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