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notes4-1 - A subspace of a vector space V is a subset H of...

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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 set of all real-valued functions defined on an interval I The set C ( I ) of all functions defined and continuous on an interval I The set C 2 ( I ) of all functions whose second derivatives are defined and continuous on an interval I
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The set P 5 of all polynomials of degree at most 5.
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Unformatted text preview: 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. Determine whether the following are subspaces of an appropriate vector space. • Span { v 1 ,..., v p } , where the vectors all belong to some vector space V • The set K of 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 • The set H of all vectors of the form 9 a-5 b 2 a 3 a + 2 b HOMEWORK: SECTION 4.1...
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