703 - For a finite subset of a vector space, the...

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1 For a finite subset of a vector space, the definitions of L.D. and L.I. are the same as those in R n . Example: S = { x 2 3 x + 2, 3 x 2 5 x , 2 x 3} of P 2 is L.D., since Example: of R 2 × 2 is L.I., since implies that a = b = c = 0. Example: S = { e t , e 2 t , e 3 t } is a L.I. subset of F ( R ), since if h ( t ) = ae t + be 2 t + ce 3 t = 0 t , then h (0) = a + b + c =0, h (0) = a + 2 b + 3 c = 0, and h ′′ (0) = a + 4 b + 9 c = 0 together imply a = b = c = 0. Example: The infinite subset {1, x , x 2 , " , x n , " } of P is L.I., since given any nonempty index set I , Σ i I c i x i = 0 implies c i = 0 for all i I . Thus its every nonempty finite subset is L.I. . Example: The infinite subset {1+ x , 1 x , 1+ x 2 , 1 x 2 , " ,1+ x n , 1 x n , " } of P is L.D., since it contains L.D. finite subsets like {1+ x , 1 x , 1+ x 2 , 1 x 2 }, in which 1(1+ x ) + 1(1 x ) + ( 1)(1+ x 2 ) + ( 1)( 1 x 2 )= 0 .
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2 Proof c 1 T ( v 1 ) + c 2 T ( v 2 ) + " + c k T ( v k ) = 0 T ( c 1 v 1 + c 2 v 2 + " + c k v k ) = 0 c 1 v 1 + c 2 v 2 + " + c k v k = 0 because T is one-to-one c 1 = c 2 = " = c k = 0. 8 Definition. A subset S of a vector space V is a basis of V if S is a L.I. set and a generating set of
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703 - For a finite subset of a vector space, the...

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