Problem 2: (2 marks)If V = R' is a vector space and let H be a subset of V and is defined asH = {(a,b,c):a² +b² = 0,c<0}. Show that H is not a subspace of vector space.%3DProblem 3: (2 marks)Let V = R’ be a vector space and let W be a subset of V, whereW = {(a,b,c):b² = c² }. Determine, whether W is a subspace of vector space ornot.Problem 4: (2 marks)Let[1221A =| 1-1and U =| - 2 , thedetermine if3-1U is in Nul A.U is in Col A.(i)(ii)

Answered: Image /qna-images/answer/20ae86af-8760-4642-8b13-a9d298390816.jpg

Answered: Problem 2: (2 marks) If V = R' is a…

Problem 2: (2 marks) If V = R' is a vector space and let H be a subset of V and is defined as H = {(a,b,c):a? +b² = 0,c <0}. Show that H is not a subspace of vector space. Problem 3: (2 marks)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Problem IV Consider the set of all vectors (x, y, z) € R³ satisfying the following condi- tions. Determine whether it is a subspace of the vector space R³ or not. Briefly explain. 61. x-y+z= 0. 62. x+y+z= 1. 63. x+y=x - 2z = 0. 64. x=y=z= 0. 65. xyz + 1 = 0. 66. x-y=y-z=z-x. 67.x=y=0. 68. y = z = 1. 69. x+y+2z ≥ 0. 70. x = y and z ≥ 0. 71. xy ≥ 0. 72. |x| = y. 73. ry20 and |x| = |y|. 74. z = x² - y². 75. z = = x² + y². 76. x² + y² + z² = 1. 77. x(x - 1) = 0. 78. x² + 2y² + 1 = 0. 79. xyz = 0. 80. xy = y = 0.
Number 7 only, please choose the letter of the correct answer
= Problem #1: Let v₁ = (7, 1, 0,-1), V2 = (0, 1, 7, 1), and b (1, 1,-1,-2). Let W be the subspace or Rª spanned by v₁ and V2. Find projwb using the Euclidean inner product.
Can you solve both questions pls
a Q.8. Let the set H =- : a² +b? >1}. Is set H a vector space? Justify your answer. Q.9. Let the set H = :ab<1 (a) Is set H a vector space? Justify your answer. (b) Is the set H close with respect to scalar multiplication? Justify your answer q Q.10. Determine if the set H of all matrices of the form is a subspace of M2. Here, p, q and 2x2 r r represent arbitrary real numbers. Q.11. Let H be the set of all vectors of the form shown below. Is H a subspace of R*? Justify your answer for each case. Here, p,q,r and s represent arbitrary real numbers. p+q-s [2p-s| p-r (a) 2р+39 р-2 (b) p+3q r+3q |1 2 3 4 5 [1 Q.12. Let A =0 1 2 3 -1 u = and v =|0 2 0 12 -2 (a) Determine if u e NulA. Could u e ColA? Justify. (b) Determine if ve CollA. Could v e NulA? Justify.
Problem 8 Let V be the set of differentiable real-valued functions with domain R. Prove that V is a subspace of the set of functions F(R, R). (You may quote anything you like from elementary calculus without proof)
Must solve all three parts. Don't solve only one or two parts. Else skip it and leave it for others to do.
4. Let V be a vector space. (a) If vo E V, is the subset W := {kvo|k e R} a subspace of V? Justify your answer. (b) If x e V and y e V, is the subset U := {sx + ty s, t e R} a subspace of V? Justify your answer.
V = {(x, y) E R² | x = y} u {(x, y) E R² | x = = -2y}. Complete the following statements to determine if V is a subspace of R. (a) V is ? If it is non-empty, give two different examples of vectors in V. If it is empty, then leave the following spaces blank. example 1: aа — example 2: b = Note: Normally, only one example is required to show V is not empty in a proof. (b) V is ? v under vector addition. If it is not closed, enter two vectors a, b E V below, whose sum is not in V. If it is closed, then leave the following spaces blank. a = b = (c) V is ? under scalar multiplication. If it is not closed, enter a scalar k and a vector c E V below, whose product is not in V. If it is closed, then leave the following spaces blank. (OO) k = and c = (d) V ? of R?.
12. Find the dimension of the subspace of R spanned by the following set of vectors. {(1, 5, 6), (2, 6, 8), (3, 7, – 1), (4, 8, 12)] (a) 1 (b) 2 (c) 3 (d) 4 (e) 0
Determine if the given Vector Subspace is LINEARLY INDEPENDENT IN THE REQUESTED VECTOR SPACE: [1 2] [5 (Item 24) }|0 1 41 [3 0] 5 ,1 3 1 in (M(3x2) [R], +;). 1 4) [4 10] l2 3. (Item 26){ sin (2x), cos(2x), sin (x) · cos(x) } in (F[0,T], R|], +;). note: (F[1, R], +,;) Vector Space of Continuous Functions over Reals : ("R"). Vector Space of Matrices of "m rows" and "n columns" over the Reais ("R"). (M[R], +,;) (m xn)