Considere as bases de beta = {(1 , 0 , 0) ; (0 ; 1 ; - 1) ; (1 ; - 1 ; 0)} .

(d) T ( x , y , z ) = ( x - z ; - z ; y + 2 z ) , determine [T]_ beta . Dica: [T]_ beta = [I]_ beta larr epsilon [T]_ epsilon [I]_ epsilon larr beta

Passo 1

Ei ei ... galera vamos terminar essa questão agora!

Seguinte precisamos de alguns dados do item (a) que serão esses logo abaixo.

[I]_epsilon larr beta = [[[1,0,1],[0,1,-,1],[0,-,1,0]]] e [I]_ beta larr epsilon = [[[1,1,1],[0,0,-,1],[0,-,1,-,1]]]

Outro caso importante notar é o vetor que foi dado para nós

T ( x , y , z ) = ( x - z ; - z ; y + 2 z )

Podemos extrair a matriz

T (x , y , z) = [[[x,0,-,z],[0,0,-,z],[0,y,2,z]]] = [[[1,0,-,1],[0,0,-,1],[0,1,2]]] * [[[x],[y],[z]]]

Assim,

T = [[[1,0,-,1],[0,0,-,1],[0,1,2]]] -> [T]_epsilon = [[[1,0,-,1],[0,0,-,1],[0,1,2]]]

Daí você se pergunta, como a matriz T é a mesma com a base epsilon ?

A resposta para essa pergunta é que a base epsilon é uma base canônica, portanto será a mesma matriz.

Depois de ter tudo em mãos galera, vamos pegar aquela dica fenomenal que a questão deu “vai ficar mamão com açúcar” (facim...facim)

Simbora para os cálculos!

Passo 2

Vejamos, queremos utilizar

[T]_ beta = [I]_ beta larr epsilon [T]_ epsilon [I]_ epsilon larr beta

Substituindo os elementos seremos capazes de obter uma multiplicação de matrizes bem simples

[T]_ beta = [[[1,1,1],[0,0,-,1],[0,-,1,-,1]]] * [[[1,0,-,1],[0,0,-,1],[0,1,2]]] * [[[1,0,1],[0,1,-,1],[0,-,1,0]]]

[T]_ beta = [[[1,*,1,+,1,*,0,+,1,*,0,1,*,0,+,1,*,0,+,1,*,1,-,1,*,1,+,1,*,(,-,1,),+,1,*,2],[1,*,0,+,1,*,0,+,1,*,0,1,*,0,+,1,*,0,+,1,*,(,-,1,),1,*,0,+,(,-,1,),*,0,+,(,-,1,),*,2],[0,-,1,-,1,*,0,+,(,-,1,),*,(,-,1,),+,(,-,1,),*,2]]] * [[[1,0,1],[0,1,-,1],[0,-,1,0]]]

[T]_ beta = [[[1,1,0],[0,-,1,-,2],[0,-,1,-,1]]] * [[[1,0,1],[0,1,-,1],[0,-,1,0]]]

Repetindo o mesmo procedimento

[T]_ beta = [[[1,*,1,+,1,*,0,+,1,*,0,1,*,0,+,1,*,1,+,0,*,(,-,1,),-,1,*,1,+,1,*,(,-,1,),+,1,*,0],[0,*,1,+,1,*,0,+,1,*,0,0,*,1,+,(,-,1,),*,1,+,(,-,2,),*,(,-,1,),0,*,1,+,(,-,1,),*,(,-,1,),+,(,-,2,),*,0],[0,*,1,+,0,*,(- 1),+,(,-,1,),*,0,0,*,0,+,(- 1),*,1,+,(,-,1,),*,(,-,1,),1,*,0,+,(,-,1,),*,(,-,1,),+,(,-,1,),*,0]]]

[T]_ beta = [[[1,1,0],[0,1,1],[0,0,1]]]

Resposta

[T]_ beta = [[[1,1,0],[0,1,1],[0,0,1]]]