Question: Let V be a finite-dimensional vector space, and let T : V -> V be a linear transformation such that T2 =T. (Such linear transformations

Let V be a finite-dimensional vector space, and let T : V -> V be a linear transformation such that T2 =T. (Such linear transformations are said to be idempotent.) (a) Show that for each v E V, we can write v as u + w for some u in im(T) and w in ker(T). (Hint: consider u = T(v) and w = v -T(v).) (b) Show that ker(T) nim(T) = {0}. (c) Show that if U is a subspace R", then the projection map proju : R" - R" is idempotent. (The definition of proju is in Section 8.1. You may assume it is a linear transformation.)

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