Which of the following sets are subspaces of the space V described?1 1[ 3 ]in V = Rº such that [7²6] [ "-6(ii) The set of all vectors V in V = R³ such that V is an eigenvector of(i) The set of all vectorsfor the eigenvalue X = -2.(iii) The set of all vectors Y(i) and (iii) only1-in V = R³ such that x + 2y + z = 3.(iv) The set of all polynomials p(t) in V = P3 = { polynomials of degree 3} such thatp'(1) = 0.(iii) only(ii) and (iii) only(ii) and (iv) onlyx(i) (ii) and (iv)=00-2110-2 -40 0 3

Check the given subsets are the subspaces of the vector space. One-step verification of the…

Answered: (ii) The set of all vectors V in V = 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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Consider the vectors in C2 (sometimes called spinors) :(cos(0/2)e-ie/2 sin(8/2)et*/2 sin(8/2)e-i$/2 ) . V2 - cos(8/2)e*#/2 (i) First show that they are normalized and orthonormal. (ii) Assume that for the vector vị is an eigenvector with the corresponding eigenvalue +1 and that the vector v2 is an eigenvector with the corresponding eigenvalue -1. Apply the spectral theorem to find the corresponding 2 x 2 matrix. (iii) Since the vectors vị and v2 form an orthonormal basis in C? we can form an orthonormal basis in C4 via the Kronecker product W11 = V1 ® V1, W12 = V1 ® v2, W21 = V2 ® V1, W22 = V2 ® V2. Assume that the eigenvalue for w11 is +1, for w12 –1, for w21 -1 and for w22 +1. Apply the spectral theorem to find the corresponding 4 x 4 matrix.
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3. If A = 0 -1 0then the bases for the eigenspaces of A2 + 21 are: 2. a) ({, = 7, v, = (1,0,0), {A2 = 4, v (-1,0,1)}, and (A = 1, v, = (-1,6,0}) b) {{ =-3, v, = (-1,6,0}), (A = 3, v (1,0,0}}, and (A = 0, v, = (1,0,0}] c) [{ = 27, v, = (1,0,0},{22 = 6, v = (-1,0,1}}, and (As = 3, s =(-1,6,0}} d) {{2 = 2, v, ={-1,0,1}}. [A2 = 23, , = [1,0,0), and (As =-1,, {-1,6,0}}
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(ii) The set of all vectors V in V = R³ such that V is an eigenvector of for the eigenvalue X = -2. (iii) The set of all vectors 0 0 •[] ar (iv) The set of all polynomials p(t) in V = P3 = { polynomials of degree 3} such that V = R³ such that x + 2y + z = 3. -2 -4 0 3