{1, x, x2, x,..., x",...}, a list of infinitely many continuous2.57 † Let Sfunctions on [a,b]. Show that if n E N and if ao, a1,.it is not possible for E-, akx* = 0, the zero function. (Remark: This exerciseshows that C[a, b] is an infinite-dimensional vector space, because it cannot havefinite dimension n for any n E N.)..., An are not all 0, thenian

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Answered: 2.57 † Let S = functions on [a,b]. Show…

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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2.53 Let Bla, b] denote the set of all bounded functions on [a, b). Is B[a, b] a vector space? Prove your conclusion.
Q.No.2 a. Show that by using Laws that if the compound statement is logically equal. [ p ^ ( ~ ( ~p V q) ) ] V (p^q) = p b. A function g: Z → Z is defined by the rule g(m,n) = m +n Determine whether it is onto or not... c. Let X= {p, q, r, s, t}, Y= {1, 23} and Z={k, e} Define one-to-one function and how many one-to-one function are there from Y to X Define onto function and how many onto function are there from Y to Z.
Exercise 3. Are the following subsets of the real line with its usual topology (in which U CR is open if and only if for every x ‹ U there is € > 0 with (x − €, x + € ) ≤ U) open, closed, both or neither? Explain. (1) [1,3); (2) (1,3) U (5,00); (3) (-∞0, ∞0); (4) { neN} U {0}; (5) The set Q of rationals. .
7. (**) Let V be a vector space over C of dimension n. Let X be the vector space of linear transformations from V to V; denote by 0 the zero map on V, which is the "zero vector" in the vector space X. Also, X contains the identity map I: V→ V. Let L: V→V be a linear transformation (so that I can be viewed as a vector in X). Further, L²= LoL EX and L³ = Lo Lo € X, etc. We define Lº := I. If q(t) is a polynomial with coefficients in C, then q(L): V → V is defined to be the following linear transformation: q(L) = apI+a₁L + a₂L²+...+ am L™ where q(t) = ao+a₁t+...+amt™ that is q(L) is just a linear combination of the powers of L, so q(L) is a "vector" in X. Assume that L 0. Prove the following: a) Using the fact that X is a vector space of dimension n² over C, show that the set of powers of L must be linearly dependent. b) Using (a), show there exists a polynomial q(t) such that q(L) = 0, the zero map. c) Show that there exists a least integer k such that there is a monic polynomial q(t) of…
8- Suppose that X,Y be a normed spaces then the function f: X → Y is continuous at a point x, EX if and only if f is a sequentially continuous at the same point. True False
Q3: Let X = (!,2) and T=}U = (1, 2-);n=l,2,u+, X %3D n+1 show that X is not T- space
12. If the null space of 0 10 0 0 1 a3. az, and if the null space of A consists of all multiples of then the value of a2 is: [a3] Page 4of3 4321
6.3) If f = u + iv is non-constant and analytic in an open set U, which one of the following are analytic as well? (a) g = u - iv, (b) h = v +iu, (c) k = -u -iv, (d) l= iu - v.
5. (*) Let V be a vector space over F and let U be a subspace of V. Define a relation ~ on V by v~wv-w€U (a) Prove that is an equivalence relation on V. Write [v] for the equivalence class of containing v € V. (b) Show that [v] = {v+u: u € U}. For u, w € V we define [v] + [w] = [v+w]~ and for λ E F we define X[v] = [v]. (c) With this definition of addition and scalar multiplication, prove that {[v]~: v € V}, the set of equivalence classes of ~, is a vector space over F. (d) Take V = R³, U = -{(+₁)=zex}. : ER and define as above. Find a set of representatives for the equivalence classes of~. (e) With V and U as in part (d), (c) tells us the set of equivalence classes of ~ is a vector space over R. What is the dimension of this vector space?
2. Let n eN and suppose X is a set with n elements. Define V as the set of all functions f : X → C. The set V together operations + and · described below Vx e X, Vc e C and Vx € X (f + g)(x) = f(x) + g(x) (c. f)(x) = c · f(x) forms a vector space (don't verify this anymore). Find a basis of the vector space V.

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