Explanation Approach: Basis of a vector space W is defined as a subset { w 1 , w 2 , ... , w n } of vectors in W that are linearly independent and span W . Normal equation is given by ( v − w * ) w i = 0 , where i = 1 , 2 , ... , n . The best least-squares approximation w * is given by: w * = ∑ a i w i a i = v T w i w i T w i Two vectors a and b are said to be orthogonal vectors if their dot product is 0 i.e. a ⋅ b = 0 Calculation: Given, W = { x : x = [ x 1 x 2 x 3 ] , x 1 + x 2 + x 3 = 0 x 1 − x 2 − x 3 = 0 } Find basis of the given vector space: x 1 + x 2 + x 3 = 0 ( 1 ) x 1 − x 2 − x 3 = 0 ( 2 ) Adding (1) and (2) ( x 1 + x 2 + x 3 ) + ( x 1 − x 2 − x 3 ) = 0 2 x 1 = 0 x 1 = 0 And x 2 + x 3 = 0 x 3 = − x 2 Thus, we can write W as W = { x : x = [ x 1 x 2 x 3 ] , x 1 + x 2 + x 3 = 0 <

Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)

5th Edition

ISBN: 9780134689531

Author: Lee Johnson, Dean Riess, Jimmy Arnold

Publisher: PEARSON

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In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied unde…

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Chapter 3.9, Problem 10E

To determine

To Find:

The basis for the subspace W. For the given vector v solve the normal equations and determine the best approximation w, and verify that v−w is orthogonal to the basis vectors for v=[3,4,0]T.

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Chapter 3.9, Problem 9E

Chapter 3.9, Problem 11E

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Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)

Ch. 3.1 - Prob. 1E Ch. 3.1 - Prob. 2E Ch. 3.1 - Exercises 1-11 refer to the vectors given in 1....Ch. 3.1 - Prob. 4E Ch. 3.1 - Exercises 1-11 refer to the vectors given in 1....Ch. 3.1 - Prob. 6E Ch. 3.1 - Exercises 1-11 refer to the vectors given in 1....Ch. 3.1 - Exercises 1-11 refer to the vectors given in 1....Ch. 3.1 - Exercises 1-11 refer to the vectors given in 1....Ch. 3.1 - Prob. 10E

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The basis for the subspace W . For the given vector v solve the normal equations and determine the best approximation w * , and verify that v − w * is orthogonal to the basis vectors for v = [ 3 , 4 , 0 ] T .

Solution Summary: The author explains the basis of a vector space W, and the best least-squares approximation for w*.

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To Find:

The basis for the subspace W. For the given vector v solve the normal equations and determine the best approximation w, and verify that v−w is orthogonal to the basis vectors for v=[3,4,0]T.

Want to see the full answer?

Chapter 3 Solutions

The basis for the subspace W . For the given vector v solve the normal equations and determine the best approximation w * , and verify that v − w * is orthogonal to the basis vectors for v = [ 3 , 4 , 0 ] T .

Solution Summary: The author explains the basis of a vector space W, and the best least-squares approximation for w*.

Concept explainers

Videos

To Find: The basis for the subspace W. For the given vector v solve the normal equations and determine the best approximation w*, and verify that v−w* is orthogonal to the basis vectors for v=[3,4,0]T.

Want to see the full answer?

Chapter 3 Solutions

To Find:

The basis for the subspace W. For the given vector v solve the normal equations and determine the best approximation w, and verify that v−w is orthogonal to the basis vectors for v=[3,4,0]T.