Linear algebra

Kenneth Hoffman

Chapter 3

Section 3.4

Exercise

9. Let [eqn: V`] be a finite-dimensional vector space over the field [eqn: F`] and let [eqn: S`] and [eqn: T`] be linear operators on [eqn: V`]. We ask: When do there exist ordered bases [eqn: BETA`]and [eqn: Beta' `] for [eqn: V`] such that [eqn: left [ S` right]_{BETA} = left [T` right]_{BETA'}``]?

proof)

We want to show that [eqn: left [ S` right]_{BETA} = left [T` right]_{BETA'}``]⇔ ∃invertible linear operate [eqn: U`] such that [eqn: T ``= U`S`U^{``-1}``]

(⇒)

By Theorem 14, [eqn: LEFT [ T` RIGHT ] _{BETA prime } = LEFT [ U` RIGHT ] _{BETA } ^{-1} ` LEFT [ T` RIGHT ] _{BETA } ` LEFT [ U` RIGHT ] _{BETA } ``]

Since [eqn: left [ S` right]_{BETA} = left [T` right]_{BETA'}``], [eqn: [S] _{B} =[U ^{prime } ] _{B} ^{-1} [T] _{B} [U ^{'} ] _{eqalign{B# }}]

It is equivalent to [eqn: S=U ^{-1} TU#].

Thus, [eqn: T ``= U`S`U^{``-1}``].

(←)

Let dimV=n.

Let [eqn: B]={[eqn: e _{1} ,e _{2} ,`...`,e _{n}]} and [eqn: B ^{'}]={[eqn: f _{1} ,f _{2} ,`...`,`f _{n}]} be ordered bases for V.

For any j=1,...,n, if [eqn: Se _{j} = sum _{i=1} ^{n} c _{i} e _{i}] , then [eqn: Tf _{j} =T(Ue _{j} )# ``````````````=`(TU)(e _{j} )# ``````````````=`(US)(e _{j} )````````````````````````````# ``````````````=`U(Se _{j} )# ``````````````=`U( sum _{i=1} ^{n} c _{i} e _{i} )# ``````````````=` sum _{i=1} ^{n} c _{i} U(e _{i} )# ``````````````= sum _{i=1} ^{n} c _{i} f _{i}]

It follows that [eqn: [S] _{BETA } =[T] _{BETA ^{'}}]

다른 순방향 풀이

(⇒)

[eqn: BETA= LEFT { {beta_i}RIGHT }] [eqn: BETA'= LEFT { {beta'_i}RIGHT }]

[eqn: a= sum _{ i=1} ^{n }a_{i}beta_{i}]

[eqn: b= sum _{ i=1} ^{n }a_{i}beta'_{i}]

그렇다면. [eqn: [S]_{BETA}[a_{i}]=[T]_{BETA'}[a_{i}]]

[eqn: [a_{i}]= pmatrix { a_1#a_2# BULLET #BULLET# BULLET#a_n}] 를 의미합니다.

[eqn: U]를 다음과 같이 정의합시다.

[eqn: U(beta_{i})=beta'_{i}]

이제 보이려는건

[eqn: S=U^{-1}TU] 입니다.

[eqn: a= sum _{ i=1} ^{n }a_{i}beta_{i}]

[eqn: b= sum _{ i=1} ^{n }a_{i}beta'_{i}]

라고 정의할 때,

[eqn: S(a)=U^{-1}TU(a)]임을 보이면 되겠죠(책이랑 반대입니다)

[eqn: pmatrix{beta _{1}&` beta _{2}&` beta _{3}&CDOTS &` beta _{n}}]는 basis의 row vector입니다.

[eqn: U ^{-1} TU(a)=U ^{-1} T(b)#~~~~~~~~~~``=U ^{-1} ( pmatrix{beta prime _{1}&` beta prime _{2}&` beta prime _{3}&CDOTS &` beta prime _{n}} [T] _{BETA prime } [a _{i} ])#```````````````````````````````````````= pmatrix{beta _{1}&` beta _{2}&` beta _{3}&CDOTS &` beta _{n}} [T] _{BETA prime } [a _{i} ] #```````````````````````````````````````= pmatrix{beta _{1}&` beta _{2}&` beta _{3}&CDOTS &` beta _{n}} [S] _{BETA } [a _{i} ]#``````````````````````````````````````=S(a)]

[eqn: pmatrix{beta _{1}&` beta _{2}&` beta _{3}&CDOTS &` beta _{n}}]는 basis의 row vector입니다.

가 됩니다.