235asign6F08

# 235asign6F08 - (a Evaluate the norm of the function f given...

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Math 235 Linear Algebra II Assignment 6 Due 09:30, October 29, 2008, in the drop box designated for your section. 1. (a) Let ( V, < , > ) be a general inner product space over the real field. The norm (length) of a vector v V is defined by || v || = the square root of < v, v > . Prove that the parallelogram law || u + v || 2 + || u - v || 2 = 2 || u || 2 + 2 || v || 2 holds for all u, v V . [Remark: This is § 6.1#24 in a general setting.] (b) Does the above law hold for a general inner product space over the complex field? Justify briefly. 2. Let V be the vector space of all real-valued continuous functions on the interval [ - 1 , 1], and let < , > be the standard inner product defined by < f, g > = Z 1 - 1 f ( t ) g (
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Unformatted text preview: (a) Evaluate the norm of the function f given by f ( t ) = 1-t for all t ∈ [-1 , 1] [see the deﬁnition of the norm given in the question above]. (b) Let g be deﬁned by g ( t ) = t for all t ∈ [-1 , 1]. Is f orthogonal to g ? 3. (a) Show that in the one-dimensional vector space C over the complex ﬁeld and with the standard inner product, the vectors u = i and v = 1 are not orthogonal to each other. (b) In the two-dimensional plane R 2 under the standard inner product, show that u = (0 , 1) and v = (1 , 0) are orthogonal to each other. 4. § 6.2 #4, 10, 12, 18 and § 6.3 #4, 8....
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