3. Prove that if r, y E RP, then ||r + yl|2 = ||x||2 + |ly||² holds if and only if r y = 0. 4. Prove that for the vector space V = C[a, b] of all continuous functions defined on the interval (a, b), the operation fg% D | f(x)g(x)dx is an inner product on V.

3. Prove that if r, y E RP, then ||r + yl|2 = ||x||2 + |ly||² holds if and only if r y = 0. 4. Prove that for the vector space V = C[a, b] of all continuous functions defined on the interval (a, b), the operation fg% D | f(x)g(x)dx is an inner product on V.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.3: Subspaces Of Vector Spaces

Problem 45E: Consider the vector spaces P0,P1,P2,...,Pn where Pk is the set of all polynomials of degree less...

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Question

3. Prove that if 1, Y E RP, then ||x + y||² = ||x||² + ||y||² holds if and only if x ·y = 0.
4. Prove that for the vector space V = C[a, b] of all continuous functions defined on the
interval [a, b), the operation f g D
| f(x)g(x)dx is an inner product on V.
a.

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