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 f and g be real-valued continuous functions in the vector space C[a, b]. Show that g(x) dx defines an inner product on C[a, b].
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.
Definition 28. Let V be a vector space over F (here F= R, C). Then the function (•₁•): V × V → F is said to be an inner product on V if the following conditions are satisfied: (x, x) is real and (x,x) ≥ 0 for all (x,x) = 0 if and only if x € 0₂. (i) (ii) (iii) (x, ay) = a(x, y) for all x, y V (iv) (x, y+z) = (x, y) + (x, z) for all (v) (x, y) = (y,x) for all x, y V. (a) (.,.): R¹ × R¹ → R is defined as xεν. (x, y) = x¹y, and for all x,y,z € V. Example: Verify that each of the following mappings define a real inner product. a EF. for all x, y ER".
7. Let F = F i + F25 + F3 k be a smooth vector field and let f(x, y, 2) be a smooth scalar function. Match the follwing af af + (a) dz ƏF3 (b) dz (c) dx dy dz F1 F2 (d) Vƒ (e) V.F (f) ▼ × F with one of (i) divF (ii) curlF (iii) grad f
1. Let f : R" → R" be a function with the property that for any v, w E R" it holds that f(v) · f(w) = V· W. I.e. f preserves the inner product on the vector space R". Show that f is a linear transfor- mation.
SHOW THAT THE FUNCTIONS SET CONTINUOUS IS A VECTOR SPACE OF CONTINUOUS FUNCTIONS IN INTERVAL I = [a, b] (F[I], +, .)
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.