3. Consider the setl' = {x = (x1, x2, 13, .. .) E C* :i=1consisting of all absolutely summable infinite sequences of complex numbers.a) Show that e is a complex vector space.b) Show thati=1defines a norm on l.

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Answered: 3. Consider the set e = {x = (r1, x2,…

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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1. Let V be the set of all ordered pairs of real numbers, and consider the following addition and multiplication operations on -() and r-(.2): +Y (M +V+1, +v +1). ki - (ku ku) Is Va vector space? Justify your answer
Which of the following polynomials from the vector space R[x] is linearly expressed by the polynomials from the subset {1+x,1-x^2,x^3)? a.) x + x^2 + x^3 b.) 1+x+x^2 c.) 1+x+x^3 d.) 1 -x^2+x^3 e.) 1+2x+x^2
Which of the following is true about the subset T = {1+ 2x - x², 2 + 3x + x²,4 + 7x − x²} of the vector space R[x]<2 of polynomials of degree 2 or less? a) T is linearly independent. b) SpanT = R[x]≤2· c) T is linearly dependent. d) None of the choices in the list is correct.
Let S {V1, V2, V3} and T = {w₁, W2, W3} be two bases in the vector space P₂ {a+bx+cx²: a, b, c = R}, where V₁ = 1+x+3x², v₂ : −2 − 2x − 7x², V3 = 1 + 2x + 6x² and = : 1 + x − x², w₂ = W1 = 1+ 2x², W3 = 3-x+5x2 Find the transition matrix from T to S. Let v = 201 - 3v2 + V3, find the coefficients (C₁, C2, C3) so that v = C₁w₁ + C₂W2 + C3W3. Hint: For the second part, you need also to find the transition matrix from S to T. =
4. Let V = {ao+aj x+a2 x² | ao, a1, a2 E R} be a vector space of polynomial up to degree three, and p(x) = pPo+P1 x+p2 x², q(x) = qo+q1 x+q2 x² E V then show that = po.4o + pP1-q1 + P2-92 defines an inner product on V.
3. Let F be a subfield of complex numbers. Let V be the vector space (over F), consisting of all polynomials, i.e. V = F[r]. Let T:V → V be the function defined by T: f(x) + (x + 1)f'(x). d For example T(x² + x) = (x + 1) . — (x² + x) = (x + 1)(2x + 1) = 2x² + 3x +1. dr (a) Is T a linear transformation? Give proper reasons. (b) Is T invertible? Give proper reasons.
3. Consider the set l' = {x = (x1, x2, x3,...) E C : i=1 consisting of all absolutely summable infinite sequences of complex numbers. a) Show that l1 is a complex vector space. b) Show that 8. ||x||1 = la| 2=1 defines a norm on l.
Let A be the set of all complex numbers z with principal argument θ ∈ [ π/4 , 5π/6 ] so that 1 ≤ |z| ≤ 2. Sketch A on the Argand plane.
Let V be an n-dimensional vector space over the field of complex numbers C. Showthat V over R has dimension 2n.
Q 06: For n≥0 , the set Pn of polynomials of degree at most n consists of all polynomials of the form p(t) = a0 + a1t + a2t2 +a3t3+......+ antn form a vector space. Where the coefficients a0, a1, a2, ..... an and the variable t are real numbers.
Consider R+ the set of all positive real numbers (here zero is not positive) equipped with the following operations: (i) u ⊕ v = uv, for any u, v ∈ R+ (ii) α ⊙ u = uα, for any α ∈ R and any u ∈ R+. show that R+ is a vector space (check the closure under the operations ⊕ and ⊙ and the 8 axioms).
= 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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3. Consider the set e = {x = (r1, x2, x3, ...) E C* : i=1 consisting of all absolutely summable infinite sequences of complex numbers. a) Show that is a complex vector space. b) Show that ||x||1 =Elei| i=1 defines a norm on l.