# 07 - VV P 18 19 20 21 23 174 11 P P 109 119 j W,W 2...

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1 1 1 109 119 P P - 174 18, 19, 20, 21, 23. P VV , 3:30-5:30 1 : ª* 1108

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2 §3 §3- 1 Ker nel and I mage V V V F V n σ α σα = = } { Im . , ) ( 7 1 1 1 1 j · ª 1 1 1 1 j · ª * σ σ Im dim : 1 Im ( ) . n V F 5 ¸® Im ; , Im , , = , = , ( ) Im ; Im . k k k α β ξ η σξ α ση β β σξ ση σ ξ η 2200 ∈ 5 + = + = + ∈ = = ∈ Q a a 1207
3 . , , ), , , ( Im ) 2 ( 1 1 1 1 1 V L n n ε ε σε = σ 1 1 1 : Im , , = = ( , , ), ; n i i i n i i n i proof V a a L α σ ξ σξ α ε σξ = = 5 ∈ = = Q M M 1 1 , , Im . n n i i i i i i b b β σε σ = = = = ∈ ∴ ⊆ ∴ = ∑ ∑ ). , , ( ) dim( 1 n rank V =

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4 . , , . ) dim(Im ) 3 ( 1 1 1 1 1 n A rankA ε ε σ = σ . ) , , ( , , , , ) , , ( ) dim(Im 1 1 1 1 1 1 1 1 1 1 A A A n n n = = σε = σ M ) 0 ( } 0 { . , ) ( : 8 1 - σ = = σα α = σ σ σ σ V Ker F V n V V V V V V V . dim 1 σ σ Ker
5 . ) ( 1 : 1 1 1 F V Ker n σ < 1 2 ( ) , , ker , 0 0 n n V F A X AX σ ε α σα < 2200 ∈ = ⇔ = & ± ε ι M ) ( r rankA = Y AX β = ← → = Q a a a 1 , , 0 , n r X X AX - = ± . ) , , ( 1 nullA Ker n ε ε = σ 1 1 2 ( , , ) ( , , , ). i n i n r X Ker L α α - = = M 1 1 ( , , )( , , ) n n r X X - = ⟨

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6 1 0 0 0 ker {( ) 0, , } V V σ β α σ α α σα - = + = + = ∈ ∈ σ n rankA n Ker Im dim ) dim( - = - = σ ) ( ) ( ], [ ) ( ], [ 1 3 x f x f x F X f x F n F n n = - σξ 1 Im [ ], , n F x Ker F - ∴ = = dim Im dim dim(Im ) 1 Ker n Ker n + = + = - 0. Ker ∴ ⇔ = 0 0 3 , , V V β σα < 2200 ∈ 5 =
7 . Im dim dim dim , ) ( 6 σ + σ =

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## This note was uploaded on 03/25/2010 for the course MATH 40 taught by Professor F.yu during the Spring '05 term at Tsinghua University.

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07 - VV P 18 19 20 21 23 174 11 P P 109 119 j W,W 2...

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