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.

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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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c) Prove that a subset W of R’ defined by W ={(x, y, z) e R² / 2x= y= 2z} is a vector space of R'.
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 T be a linear operator on a real vector space V , and define f : V × V → R byf(x, y) = (x, T y) for all x, y ∈ V .(a) Prove that f is a bilinear form.(b) Prove that f is symmetric if and only if T is self-adjoint.(c) What properties T should have for f to be an inner product on V ?(d) Explain why f may fail to be a bilinear form if V is a complex inner product space.
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.
7. Let V(C) be the vector space of all continuous complex valued functions on 1 [0, 1]. If f(t), g (1) e V then show that (f (t), g (t)) = [ƒ (t) g (t) dt defines an inner product on V (C).
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, T:R → R';T(x, y, z) = (x – y,2x + 3y+ 2z,x+ y+2z) 7. verify that [T],[v]s = [T (v)]s for any vector v in R'.
(5) Fix a 1-cell o: I R2. Consider the operation that takes each 1-form w EN(R²) to the integral faw. (a) Is this a norm on the vector space Q'(R²) = { ƒ dx +g dy : f.g: R → R }? Prove or explain why not. (b) Is the absolute value w w/ a norm? Prove or explain why not. (Hint: One of the norm axioms says |v|| = 0 if and only if v = 0.) 1
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.