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C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes4-1

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SECTION 4.1 VECTOR SPACES AND SUBSPACES A vector space is any nonempty collection V of objects that can be added and multiplied by scalars (numbers) so that the resulting algebra is nice, that is, like the algebra of R n . The objects in V are called vectors . A precise list of the nice algebraic properties is in the definition on page 217, supplemented by the simple facts in the blue box. EXAMPLES. R n itself The space S of all infinite sequences of numbers { z k } = ( z 0 , z 1 , z 2 , . . . ) The space P 5 of all polynomials of degree at most 5.

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The space C ([0 , 1]) of all continuous functions with domain [0
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Unformatted text preview: , 1]. A subspace of a vector space V is a subset H of V such that 1. the zero vector is in H , 2. the sum of two vectors in H is again in H , and 3. any scalar times any vector in H is again in H . EXAMPLES. • Span { v 1 , . . . , v p } • All polynomials of degree at most 5 whose value at 1 is 0. • All vectors of the form 3 2 a-b 3 a + 2 b • All vectors of the form 9 a-5 b 2 a 3 a + 2 b HOMEWORK: SECTION 4.1...
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