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 theinterval [a, b), the operation f g D| f(x)g(x)dx is an inner product on V.a.

Answered: Image /qna-images/answer/1d90fd04-f639-4cab-8667-e1b07dcc2b52.jpg

Answered: 3. Prove that if r, y E RP, then ||r +…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let V be the vector space of polynomials in & over R of degree <3 and define an inner product f. g = f₁¹ f(x)g(x)dxæ for all f, g € V. You may assume that this makes V into an inner product space. Define the linear map a: V→ V by a(f) (x) = f(x)+xƒ'(x) where the f' denotes differentiation. Which of the following holds with respect to this inner product? Select one: O a is self-adjoint O None of the others apply Ⓒa* = I-a, where I denotes the identity map O a is `anti-self adjoint' meaning a* = -a a is orthogonal
Let C₁ be the directed line segment that starts at (2,0) and ends at (−3,−5) and C₂ be the parabola with parametrization x = t, y = 4 — t² for t € [−3, 2]. Let F be the vector field F(x, y) = (x² − 4y) î + (2x + 3y²) ĵ. (a) Evaluate Sc₁ F· dr. F. (b) Evaluate fF.dr where C' is the positively oriented path consisting of the parabola y = 4-x² starting from (2, 0) to (−3, −5) and the line segment from (−3,−5) back to (2,0). (c) Use the results in (a) and (b) to evaluate Sc₂ F · dr.
LetV = R[x]3, W = R[x]6 be the vector spaces consisting of zero and polynomials in x of degree ≤ 3 and degree ≤ 6 respectively. This question concerns the map α : V → W which sends a polynomial f(x) in V to f(x^2) in W. (a) Show that α is linear as a map V →W. (b) Find the nullity ν(α) and the rank ρ(α). (c) Let U = Image(α) ⊆ W and find a subspace Z ⊆ W such that W = U ⊕ Z.
Let V be the vector space of polynomials in over R of degree ≤ 3 and define an inner product f . g = f¹ f(x)g(x)dx for all f, g € V. You may assume that this makes V into an inner product space. Define the linear mapa: V → V by a(ƒ)(x) = f(x) + xƒ'(x) where the f' denotes differentiation. Which of the following holds with respect to this inner product? Select one: O O O a is self-adjoint a is orthogonal a is 'anti-self adjoint' meaning a* = -α a = I-a, where I denotes the identity map None of the others apply
44. Products of scalar and vector functions Suppose that the sca- lar function u(t) and the vector function r(t) are both defined for a

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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.