LectureNotes_7.3and7.4.pdf

Suppose a null space what ab nwllj ja fre n two

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Suppose A null space? What ab ,nwLL;j-.J(A) fre" n> +. two-dimensional ? tran\<cA) < q n,rLli! (A) >3. Note. Ax - 0 has only the trivial so lution (* nullity(A) - 0 <+ '_**- <-:? nuj"L\ecL CA\ = Z-6 \ n,o tsn*'t c}$fl\ o 6 (' rank(A) -n (1}+ )
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_> \ C o \ 7 -3 t *rt r -3c+g-rs Z L*t -3S-{s-q-5*3 o *3 *t z 5 -'z -s 2 Z f) e E z_te _,q 3 --\ L) l) (" :G e t LOP 13 a ,# .+:::-# I -= e*) :8r
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The Fundarnental Theorern of Invertible Vlatrices Theorern (The Fundamental Theorem of Invertible l\{atrices: VS) If A 'is an n x n matrir, then the follow,ing statements a,re equ,iualent: (") A i,s 'inuertzble. (b) The reduced rou) echelon form of A ,is In. (.) A can be written as a product of elementary matrices. (d) Ax - b has a un'ique solut,ion for all b IR,. (") Ax - O has only the triu'ial solut,iou x - 0; (no nontriu,ial solut,ions). (f) The column uectors of A are l'inearly independent. (g) The column uectors of A spa,n lR'. (h) The column uectors of A form a bas,is /or R' . (i) Tlt e ro'ur uectors of A are l,inearly ,independent. (j) The rou) uectors of A spa,n IR'. (k) The rou) uectors of A f orm a bas,is /or IR' . (r) det(A) + o. ) ^ - 0 'is not an e'igenualue of A. ) Ta 'is one-to-one. ) fa ,is onto. ) rank( A) : ,. ( m n o p - -? ( ( ( ( 7 q) nullity( A) : O.
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Orthogonal Complements Definition If S is a non-empty set in R', then the orthogonal cornplement of S, denoted by St, is the set of all vectors in IR' that are orthogonal to every vector in S. That is, Sr - {x R.' : x.v - 0 for all v ,S k cnccr k I nrr-\\ CA) = 1i (A) 2 . ..,?*,D] t ( >ee- s\I.Xe- q) Erample. Find 5r if ,S - {(0, 1,6), (0,0,2)}. Theorern (7.3.3) If S 'is a non-empty set'in R', then SL 'is a subspace o/R'. Proof 8 see- Te.^t f". f""4 (
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cf} "{ ( ,t t I (r ; C l'\ fi ( t I 1 1. n ,l \_u 0 el J 'l,2 J rr) ,l ti 1 q ( tx t l \ $ ,l
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i1 Properties of Orthogonal Complements Theorem (7.3.4) (u) If W ,is a subspace o/R' , then WL nW - {0}. (b) If S ds a non-empty subset o/R' , thensr - span(,S), .+ orlhc'O (.) If W 'is a subspace o/ R' , then (W')' - W . or .f,ne\.,ne,ct- f rJ.
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