Chapter 3 Linear transformations

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Remarks 3.3.7(1) If A is similar to B, thendet(A)=det(B)rank(A)=rank(B)nullity(A)=nullity(B)tr(A)=tr(B)(2) If A and A' are two matrices of a linear operator T:V → V with respect to twobases of V, then A and A' are similar.Exercise 3.31. Prove if A is similar to B then(a) det(A)=det(B);(b) rank(A)=rank(B);(c) tr(A)=tr(B).2. Prove if A is similar to B and A is invertible, then B
is invertible and A −1is similar to B − 1 .3. Let⎡⎢=⎢⎢⎣b⎤⎥⎥⎥⎦⎡⎢=⎢⎢⎣c⎤⎥⎥⎥⎦⎡⎢=⎢⎢⎣A c a b , B a b c , C b c a .acbacbacbaShow that A,B and C are similar to each other.4. Prove that the following two diagonal matrices are similar if and only if b1, b2,...,bnisa re-arrangement ofa ,...,1, a2a n,acbacb⎤⎥⎥⎥⎦a⎡⎢⎢⎢⎢⎣10a2..0ab⎤ ⎡⎥ ⎢⎥,⎢⎥ ⎢⎥ ⎢⎦ ⎣10b...0nb n2⎤⎥⎥⎥⎥⎦.3.4 Invariant subspaces ( optional )Given a linear operator T on V, we wish to find a basis B of V such that [T] Bis in simplerform, such as a diagonal matrix. This problem is closely related to invariant subspaceproblem.Definition 3.4.1 Let T:V → V be a linear operator. A subspace W of V is said to beinvariant under T ifT(W) ⊆ W. Then W is called W an invariant subspace of T.Example 3.4.2{0} and V are invariant under any T. These are called the trivial invariant subspaces .
Page 1 and 2: The Landmark Tshipi Acquisition & C
Page 3 and 4: Thus P 2 is isomorphic to R 3 , and
Page 5 and 6: ker(T) ={p(x): p’(x)=0}={ all con
Page 7 and 8: Also (1,0 )=T(1,0,0)=T(e 1 ), (1,1)
Page 9 and 10: Thus [T]S, B=A=[ F(e 1) F(e 2) …
Page 11: Solution: By the above resultP=[[1]

Chapter 3 Linear transformations

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