SOLUTION: Let T be a linear operator in V. Prove that T^2 = 0 if and only if the range T is a subset of null T.

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Question 15125: Let T be a linear operator in V. Prove that T^2 = 0 if and only if the range T is a subset of null T.
Answer by khwang(438) About Me  (Show Source):
You can put this solution on YOUR website!
Let T be a linear operator in V. Prove that T^2 = 0 if and only if the range T is a subset of null T.
[This question is very easy,you should know how to prove it)
Proof: (==>)
If y is in R(T) (range of T) then there is x in V such that
y = T(x). Hence, T%28y%29+=+T%28T%28x%29%29+=+T%5E2%28x%29+=+0+
This shows R(T) is a subset of N(T).
(<==)
If R(T) < N(T), then
for any x in V, T(x) is in R(T) < N(T).
Hence , +T%5E2%28x%29+=+T%28T%28x%29%29+=+0+
QED

You should work hard.

Kenny