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...
View
Full Document
 Spring '08
 Staff
 Linear Algebra, Algebra, Transformations, Vector Space, nonzero linear transformation

Click to edit the document details