Section 5.4: 15: To prove the Cayley-Hamilton theorem for matrices, we can use the isomorphism between n × n matrices M n × n ( F ) and linear transformations L ( F n ) given by A 7→ L A . The key additional fact that we need about this isomorphism, is that it also preserves multiplication, and thus a simple induction shows that for any polynomial g ( t ) ∈ P ( F ), g ( L A ) = L g ( A ) . We also know, by deﬁnition, that the characteristic polynomial f ( t ) of A and of L A are the same. Thus by Cayley-Hamilton for linear operators, we know 0 = f ( L A ) = L f ( A ) , and since the map A 7→ L A is one-to-one, we get f ( A ) = 0. Section 5.4: 20: If V is T cyclic with generator v0 , and if U commutes with T , then we can write U ( v0 ) = g ( T )( v0 ) for some polynomial g ( t ) (Problem 13). Then since U and T commute, we know by a simple induction that U and T j also commute so we can write U ( T j ( v0 )) = T j ( U ( v0 )) = T j ( g ( T ) v0 ), but g ( T ) also commutes with T j so we get U ( T j ( v0 )) = T j ( g ( T ) v0 ) = g ( T )( T j v0 ). Since the set of vectors T j v0 spans V , this implies that Uv = g ( T ) v for any vector v ∈ V
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