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**Unformatted text preview: **U is also a nonzero linear transformation, we must have b = 0. Sec. 2.3 #11: Let V be a vector space, and let T : V V be linear. Prove that T 2 = T if and only if R ( T ) N ( T ). Proof. = Assume T 2 = T . We must show R ( T ) N ( T ). So let x R ( T ). Then x = T ( y ) for some y V . Then T ( x ) = T ( T ( y )) = T 2 ( y ) = T ( y ) = 0 , which shows that x N ( T ). = Conversely, assume R ( T ) N ( T ). Let v V be arbitrary. We must show T 2 ( v ) = 0. But T 2 ( v ) = T ( T ( v )), and T ( v ) R ( T ). So R ( T ) N ( T ) implies T ( v ) N ( T ). So T ( T ( v )) = 0, i.e. T 2 ( v ) = 0. 1...

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