2.2 Matrix - 2.2. 2.2. The matrix representation of a...

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2.2. 2.2. The matrix representation of a linear transformation 1. β = { v 1 ,v 2 , ··· ,v n } , γ = { w 1 ,w 2 , ··· ,w m } bases for V and W, respectively (a) T (p.82 Theorem 2.7(a)) (b) T (p.73 The corollary to Theorem 2.6 and p.80) (p.80) Let T:V W is linear. Then for each j , 1 j n , there exist unique scalars a ij ,b ij F, 1 i m s.t T( v j ) = m i =1 a ij w i , U( v j ) = m i =1 b ij w i for 1 j n Suppose [ T ] γ β = ( a ij ) m × n ,[ U ] γ β = ( b ij ) m × n If [ T ] γ β = [ U ] γ β , then T( v j ) = m i =1 a ij w i = m i =1 b ij w i =U( v j ) for all a ij ,b ij F, v j β . Hence T=U (c) F ([ T ] γ β is an n × m matrix ) (d) T (p.83 Theorem 2.8 (a)) (e) T (0 ∈ L ( V,W )) and Theorem 2.7 (a)) (f) F (p.104 L ( V,W ) = M m × n ( F ), L ( W,V ) = M n × m ( F )) ( cf ) ( L ( V,W ) = L ( W,V )) but ( L ( V,W ) 6 = L ( W,V )) 2. Compute [ T ] γ β (a) T(1,0)=(2,3,1)=2 w 1 + 3 w 2 + 1 w 3 , T(0,1)=( - 1 , 4 , 0) = - 1 w 1 + 4 w 2 + 0 w 3 [ T ] γ β = 2 - 1 3 4 1 0 77 PNU-MATH
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2.2. (b) T(1,0,0)=(2,1), T(0,1,0)=(3,0), T(0 , 0 , 1) = ( - 1 , 1) [ T ] γ β = ± 2 3 - 1 1 0 1 (c) [ T ] γ β = (1 , 0 , - 3) (d) [ T ] γ β = 0 2 1 - 1 4 5 1 0 1 (e) [ T ] γ β = 1 0 ··· 0 1 0 ··· 0 . . . . . . ··· . . . 1 0 ··· 0 (f) [ T ] γ β = 0 0 ··· 0 1 0 0 ··· 1 0 . . . . . . ··· . . . 0 1 ··· 0 0 1 0 ··· 0 0 (g) [ T ] γ β = (1 , 0 , ··· , 0 , 1) 3. (a) T(1,0)=(1,1,2)= - 1 3 (1 , 1 , 0) + 0(0 , 1 , 1) + 2 3 (2 , 2 , 3) T(0,1)=( - 1 , 0 , 1)= - 1(1 , 1 , 0) + 1(0 , 1 , 1) + 0(2 , 2 , 3) [ T ] γ β = - 1 3 - 1 0 1 2 3 0 (b) T(1,2)=( - 1 , 1 , 4)= - 7 3 (1 , 1 , 0) + 2(0 , 1 , 1) + 2 3 (2 , 2 , 3) T(2,3)=( - 1 , 2 , 7)= - 11 3 (1 , 1 , 0) + 3(0 , 1 , 1) + 4 3 (2 , 2 , 3) 78 PNU-MATH
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2.2. [ T
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2.2 Matrix - 2.2. 2.2. The matrix representation of a...

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