3. What it means to be well-defined:(a) Let C(0, 1]) be the vector space of complex-valued continuous functions onDirac function oo : C([0, 1]) -> C by0, 1]. Define thedo(f f(0)Prove that for all a, ßE R and f,g E C((0,1]),we haveo(afBg)ado(f)+ Bo(g)C(0, 1)(In other words, 60 is a linear functional on(b) Let R(0, 1]) be the vector space of Riemann-integrable functions ontransformation? Explain your answer0, 1. Is o also a linearcarefully.

The problem concerns well-definedness, linear functionals and delta functional .

Answered: 3. What it means to be well-defined:…

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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can you help me solve part b and c please?
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
None
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].
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 Pn be the vector space of polynomials of degree at most n and define L:P2 → P1 : p(t) → p'(t), p(t) e P2. (a) Show that L is linear. (b) Compute ker(L) and find dim(im(L)). (c) Is L is injective? Is L surjective? Justify your answers.
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.
3. Let V denote the vector space of all functions ƒ : R → R, equipped with addition + : V × V → V defined via (ƒ + g)(x) = f(x) + g(x), x € R, and scalar multiplication · : R × V → V defined via (\ · ƒ)(x) = \ƒ(x), xERn Now let W = {ƒ : R → R : ƒ(x) = ax + b for some a, b ≤ R}, i.e. the space of all linear functions R → R. (a) Show that W is a subspace of V. (You may assume that V is a vector space). (b) Find a basis for W. You should prove that it is indeed a basis.
Let V be the vector space of continuous functions and the scalar product in V is defined as (f,g)=L f(x)g(x)dx , then find the norm of vector f(x)=x+1 ? a. 2 b. 4 c. V3 d. 3
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
= 4. Let P₂ be the vector space of real polynomials of degree at most 2, that is P₂ = {ao + a₁ + a₂x²2 a, ER}. Consider the inner product on P₂ defined by Define T: P2 P₂ by (p\q) = p(x)q(x) dx. 0 T(ao + ax + a2x²) = a₁x. (a) Show that T is not Hermitian (i.e. self-adjoint). (b) Consider the basis B = {1, x, x2} for P₂. Show that the matrix [T]g is Hermitian. Explain why this does not contradict with result from Part (a).

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