 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2023/digital-systems-linear-algebra-and-introduction-to-formal-proof/5/b/solution

Take

V = R^{2}

h (x, y) = (y, 0)

. Then

R (h) = N (h) = span {(1, 0)} .

Their intersection is nonzero and their sum is only one-dimensional. Hence $V \neq = R (h) \oplus N (h)$ . This nilpotent linear map has

h^{2} = 0

, illustrating why taking sufficiently high powers changes the situation.

 New to topics? Read the docs here!