Answered: Q1) Use Gram-Schmidt orthonormalization… | bartleby

Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CM: Cumulative Review

Problem 16CM

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Q1) Use Gram-Schmidt orthonormalization process to transform the
basis B = {(1,0, 0}, (1, 1, 1), (1,1, –1)} for R" into an orthonormal
basis.

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Suppose the matrix A is used to transform points in the plane iteratively. That is, given a point v, consider the sequence vn = Anv. Letting U = [u1 u2] so that ui is an eigenvector associated to λi and letting v = c1u1 + c2u2 what is a simple expressions for an and bn so that vn = Anv = anu1 + bnu2.
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Suppose R4 has the Euclidean inner product. Apply the Gram Schmidt process to transform the base {u1, u2, u3, u4} into an orthonormal base. Where u1 = (0, 2, 1, 0), u2 = (1, −1, 0, 0), u3 = (1, 2, 0, −1) and u4 = (1, 0, 0, 1). Note: Do not skip any step to arrive at the result, apply the Gram Schmidt to arrive at the result (In the image the enunicoado is better seen)
Suppose the matrix A is used to transform points in the plane iteratively. That is, given a point v, consider the sequence vn = Anv. Letting U = [u1 u2] so that ui is an eigenvector associated to λi and letting v = c1u1 + c2u2 what is a simple expressions for an and bn so that vn = Anv = anu1 + bnu2. If A and B are similar square matrices and say S witnesses this, that is A = SBS−1 , show that if (λ, v) is an eigenvalue/eigenvector pair for A, then (λ, S−1v) is an eigenvalue/eigenvector pair for B.
use gaussian and gauss jordan elimination or LU-decomposition
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Let A be an n × n matrix with distinct real eigenvaluesλ1, λ2, . . . , λn. Let λ be a scalar that is notan eigenvalue of A and let B = (A − λI)−1. Showthat if the power method is applied to B, then the sequence of vectors will converge to an eigenvector of A belonging to the eigenvalue that is closest to λ.

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