Explanation Given: A square matrix of 3 dimensions, such that is formed over the perpendicular column vectors, solution of the equation is A X = b and can be written as x 1 A 1 + x 2 A 2 + x 3 A 3 = b Where A = [ A 1 , A 2 , A 3 ] , each column is non-zero and that A 1 ⋅ A 2 = 0 , A 1 ⋅ A 3 = 0 and A 3 A 2 = 0 . Approach: Using orthogonality of the columns for finding the variables and dot product to solve for the query, such that A = [ A 1 , A 2 , A 3 ] , each column is non-zero and that A 1 ⋅ A 2 = 0 , A 1 ⋅ A 3 = 0 and A 3 A 2 = 0 , and by given solution over x 1 A 1 + x 2 A 2 + x 3 A 3 = b Calculation: Given that A is the 3 dimensional square matrix and can be represented in column vectors as A = ( A 1 , A 2 , A 3 ) Where A 1 ⋅ A 2 = 0 A 1 ⋅ A 3 = 0 A 2 ⋅ A 3 = 0 If b is the three space vector, b = [ b 1 b 2 b 3 ] Then the equation A X = b Can be written as x 1 A 1 + x 2 A 2 + x 3 A 3 = b By multiply the following A 1 Thus ( x 1 A 1 + x 2 A 2 + x 3 A 3 ) ⋅ A 1 = b ⋅ A 1 ( x 1 A 1 A 1 + x 2 A 2 A 1 + x 3 A 3 A 1 ) = b ⋅ A 1 And by the orthogonality condition A 1 ⋅ A 3 = 0 A 2 ⋅ A 3 = 0 So, x 1 ( A 1 ⋅ A 1 ) = x 1 [ ( a 11 ) 2 + ( a 21 ) 2 + (

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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Chapter 2.CE, Problem 6CE

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Let A be a (3×3) matrix, given in column A=[A1,A2,A3], by using orthogonality of the vectors, determine the formula for x1,x2 and x3.

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

Ch. 2.1 - In Exercises 1-4, graph the geometric vector u=AB...Ch. 2.1 - Prob. 2E Ch. 2.1 - Prob. 3E Ch. 2.1 - Prob. 4E Ch. 2.1 - Let u=AB and v=CD where...Ch. 2.1 - In Exercises 6-9, find the unspecified coordinates...Ch. 2.1 - In Exercises 6-9, find the unspecified coordinates...Ch. 2.1 - In Exercises 6-9, find the unspecified coordinates...Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E

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Let A be a ( 3 × 3 ) matrix, given in column A = [ A 1 , A 2 , A 3 ] , by using orthogonality of the vectors , determine the formula for x 1 , x 2 and x 3 .

Solution Summary: The author explains how to determine the formula for x_1,2 based on orthogonality of the vectors.

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Let A be a (3×3) matrix, given in column A=[A1,A2,A3], by using orthogonality of the vectors, determine the formula for x1,x2 and x3.

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