Let V be a vector space, and let T: V-> V be linear. Prove that T^2 = T0 if and only if R(T) is contained in N(T).

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- October 11th 2008, 11:19 AMleungstaComposition of Linear Transformations (2)
Let V be a vector space, and let T: V-> V be linear. Prove that T^2 = T0 if and only if R(T) is contained in N(T).

Thanks.. - October 11th 2008, 12:49 PMwatchmath
Suppose

Let . Then for some x. Now . This shows that .

Conversely let .

for every x, which implies that . Hence