571_3.3 - 3.3 LINEAR INDEPENDENCE KIAM HEONG KWA p. 128 Let...

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Unformatted text preview: 3.3 LINEAR INDEPENDENCE KIAM HEONG KWA p. 128 Let V be a linear space and let u , v k V , k = 1 , 2 , ,n . Then the statements (1) u Span( v 1 , v 2 , , v n ) ; (2) Span( u , v 1 , v 2 , , v n ) = Span( v 1 , v 2 , , v n ) . are equivalent. Proof of (1) (2). Statement (1) means that u = n k =1 b k v k for some scalars b k s, k = 1 , 2 , ,n . Hence for any scalars c l s, l = 0 , 1 , ,n , c u + n l =1 c l v l = c n k =1 b k v k + n l =1 c l v l = n k =1 c b k v k + n k =1 c k v k = n k =1 ( c b k + c k ) v k Span( v 1 , v 2 , , v n ) . This shows that Span( u , v 1 , v 2 , , v n ) Span( v 1 , v 2 , , v n ) . Since the inclusion Span( v 1 , v 2 , , v n ) Span( u , v 1 , v 2 , , v n ) is clear, we have shown that (1) implies (2). Proof of (2) (1). Since u = 1 u Span( u , v 1 , v 2 , , v n ) , (2) implies (1) immediately. p. 129 Let V be a linear space and let v k V , k = 1 , 2 , ,n . The vectors v 1 , v 2 , , v n are said to be linearly dependent if there is Linear depen- dence a c = ( c 1 ,c 2 , ,c n ) T R n , c 6 = , such that n X k =1 c k v k = c 1 v 1 + c 2 v 2 + + c n v n = . p.129 Let V be a linear space and let v k V , k = 1 , 2 , ,n . The vectors v 1 , v 2 , , v n are said to be linearly independent if for a Linear indepen- dence c = ( c 1 ,c 2 , ,c n ) T R n , the condition n X k =1 c k v k = c 1 v 1 + c 2 v 2 + + c n v n = implies that c = . Date : June 27, 2011. 1 2 KIAM HEONG KWA p. 133 Let S = { v 1 , v 2 , , v n } V , where V is a linear space. An Theorem 3.3.2 element v Span( S ) can be written uniquely as a linear combination of S if and only if S is a linearly independent set. Proof. Let v = n X k =1 b k v k = n X k =1 c k v k , where b k s and c k s are scalars, be two representation of v as linear combinations of S . Note that n X k =1 ( b k- c k ) v k = n X k =1 b k v k- n X k =1 c k v k = v- v = ....
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This note was uploaded on 11/11/2011 for the course MATH 571 taught by Professor Kim during the Summer '08 term at Ohio State.

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571_3.3 - 3.3 LINEAR INDEPENDENCE KIAM HEONG KWA p. 128 Let...

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