document.write( "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. \n" ); document.write( "

Algebra.Com's Answer #7529 by khwang(438) $\"\"$ $\"About$

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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.
\n" ); document.write( " [This question is very easy,you should know how to prove it)
\n" ); document.write( " Proof: (==>)
\n" ); document.write( " If y is in R(T) (range of T) then there is x in V such that
\n" ); document.write( " y = T(x). Hence, $\"T%28y%29+=+T%28T%28x%29%29+=+T%5E2%28x%29+=+0+\"$
\n" ); document.write( " This shows R(T) is a subset of N(T).\r
\n" ); document.write( "\n" ); document.write( " (<==)
\n" ); document.write( " If R(T) < N(T), then
\n" ); document.write( " for any x in V, T(x) is in R(T) < N(T).
\n" ); document.write( " Hence , $\"+T%5E2%28x%29+=+T%28T%28x%29%29+=+0+\"$
\n" ); document.write( " QED\r
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\n" ); document.write( "\n" ); document.write( " You should work hard.\r
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\n" ); document.write( "\n" ); document.write( " Kenny\r
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