Let W be the subspace of R* spanned by the vectorsuj =(1,0, 1,0), u2 = (0, 1, 1,0), and uz = (0,0, 1,–1).Use the Gram-Schmidt process to transform the basis {u1, u2, u3} into an orthonormal basis.

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Answered: Let W be the subspace of R* spanned by…

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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Let W C R" be a subspace and let W- C R" be its orthogonal complement. Let S {u1,..., uk} be an orthogonal basis for W and S' The goal of these exercises is to give two different arguments for the fact that {V1,..., Vi} be an othogonal basis for W-. (*) dim W + dim W- = n. 4. Show that SUS' is an orthogonal basis for R" and deduce (*).
Consider a physical system whose three-dimensional state space is spanned by the orthonormal basis formed by the three kets {|e1>, |e2>, |e3>}. In the basis of these three vectors, taken in this order, the Hamiltonian H^ and the two operators B^ and D are defined by: H = ħwo 3 i 0 i 30 0 02 7 B÷bo -i i 1- i 1+ 6 | (0)) = 7 1+i 1 - i (e₁] (0)) (€₂ (0)) (€3) (0) 0 0 2α where wo and bo are constants. Also using this ordered basis, the initial state of the system is given by: = D 0 2α 2a 0 -3a Suppose that the initial state (0)> was left to evolve until t = 0. Q: After measuring operator Ô at t#0 and then B, what is the probability of finding the system in ground state?
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Apply the Gram-Schmidt orthonormalization process to transform the given basis for R" into an orthonormal basis. Use the vectors in the order in which they are given. B = {(3, 1, 0), (0, o, 3), (1, 1, 1)} u = uz = Uz =
Use the Gram-Schmidtorthonormalization process to find an orthonormal basis of ?2(with the dot product)from the basisthe basis{(4,−3,0),(1,2,0),(0,0,4)}.
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Let W be the subspace of R* spanned by the vectors uj = (1,0,1,0), u2 = (0, 1, 1,0), and uz = (0,0, 1, –1). Use the Gram-Schmidt process to transform the basis {u1, u2, u3} into an orthonormal basis.