4. Which of the following subsets of R³ constitute a subspace of R³?[Here, x = (₁, §2, §3).](a) All x with §₁ = ₂ and §3 = 0.(b) All x with ₁ =-₂ + 1.(c) All x with positive1, 2, §3.(d) All x with ≤₁ − ₂ + 3 = k = const.

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Answered: 4. Which of the following subsets of R³…

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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6.2.5
1. Find curl(v) of the following in terms of (x1, x2, x3) and (e1, e2, e3). If curl(v) = 0, find the function f such that Vf = v. a. v(x) = (x² + x + x3)-'(x1e1 + x2€2 + X3@3). b. v(x) = (xỉ + X2X3)e1 + (x3 + X1X3)e2 + (xž + x2X1)e3 c. v(x) = e"1(sin(x2) cos(x3)e1 + sin(x2) sin(x3)e2 + cos(x2)e3)
7. Let A, B C R be the sets (-x∞, –1) U [0, 1) U [2, 0) and B = [-2,0.5) U [0.5, 4]. Write %3D down the sets (i) AC = R \ A, (ii) BC = R \ B, and (iii) AN B. %3D
8. Decide which of the following sets are linearly independent in R. Justify your answer in each case. (a) X₁ = {(1,0), (0, 1)} CR² (b) X₂ = {(1,0), (2,0)} CR² (c) X3 = {(-1,0), (0,0)} CR² (d) X₁ = {(1,0), (0, 1), (1, 1)} CR² (e) X5 = {(1,0,0), (0, 1, 0), (1, 1, 1)} CR³ (f) X6 = {(1,0,0), (0, 1, 0), (1, 1, 1), (-1,0,1)} CR³ (g) X7 = {(0, 0, 0), (0, 1, 0), (1, 1, 1), (-1,0,1)} CR³ (h) Xs = {(0, 1, 0), (1,1,1)} CR³ (i) X9 = {(4,3,0,0), (0, 0, 1, 1), (0, 0, 0, 1), (1, 0, 0, 1), (0, 1, 0, 1)} CR¹ (j) X10 = {(1,2,0,0), (0, 2, 3,0), (0, 0, -1, 1)} CR¹
2. (a) Show that the polynomials p, (x)=1+x, p,(x)=1+2.x – x², p;(x)=x+x² are linearly independent. (b) Decide if polynomial q(x)=-1+x is in the span{p, (x), P2 (x), P;(x)} .
5. Let u = [4] and v= [2] find a and b such that au+bv = 0 [8]
Find x1, x2, x3 with gauss jordan
2 2 3. Evaluate f(x²+y²-ixy )dz from 1-i to 2+i2 along the following paths (a) horizontally from 1-i to 2-i then vertically to 2+i2 (b) directly on the line from 1-i to 2+i2
from ( 0,1) to (2,5)**
3. (.) Denote the number of derangements of {1, 2,..., n} by Dn. Using the formula for the number of the derangement that we found in class, provide a proof of the relation D, = (n – 1)(D-2 + Dn-1); (n = 3,4, 5, ...).

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4. Which of the following subsets of R³ constitute a subspace of R³? [Here, x = (₁, §2, §3).] (a) All x with §₁ = ₂ and §3 = 0. (b) All x with ₁ = - ₂ + 1. (c) All x with positive 1, 2, §3. (d) All x with ≤₁ − ₂ + 3 = k = const.