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3. Let (*) be the following system of five linear equations in three unknowns x1, x2, X3 over R: 2x1 + x₂ + x3 = b₁ 3x1 + x₂ + x3 b₂ x1 + x2 + x3 = b3 4x1 + x2 + 2x3 = b4 -9x1 - 3x2 - 5x3 = b5. (a) Show that the system (*) has a solution if and only if -264 - b5. b3 = 2b1 b2 = = (b) Suppose (b₁,b2, b3, b4, b5) (-1, 4, 2, 4, 3). Find the corresponding normal equation of the system (*), and use it to find all least squares solutions of (*). You may use a calculator for this part. = (*) (c) Again suppose (b₁,b2, b3, b4, b5) (−1, 4, 2, 4, 3). Instead of normal equations, now use orthogonal projection to find all least squares solutions s of (*) and the error E= ||b- As||2. Note that the necessary computation has been done in Homework 8 Problem 4(c), so you should use the result there and do this part by hand. What is the geometric meaning of the error E in the context of Homework 8 Problem 4(c)? -

3. Let (*) be the following system of five linear equations in three unknowns x1, x2, X3 over R: 2x1 + x₂ + x3 = b₁ 3x1 + x₂ + x3 b₂ x1 + x2 + x3 = b3 4x1 + x2 + 2x3 = b4 -9x1 - 3x2 - 5x3 = b5. (a) Show that the system (*) has a solution if and only if -264 - b5. b3 = 2b1 b2 = = (b) Suppose (b₁,b2, b3, b4, b5) (-1, 4, 2, 4, 3). Find the corresponding normal equation of the system (*), and use it to find all least squares solutions of (*). You may use a calculator for this part. = (*) (c) Again suppose (b₁,b2, b3, b4, b5) (−1, 4, 2, 4, 3). Instead of normal equations, now use orthogonal projection to find all least squares solutions s of (*) and the error E= ||b- As||2. Note that the necessary computation has been done in Homework 8 Problem 4(c), so you should use the result there and do this part by hand. What is the geometric meaning of the error E in the context of Homework 8 Problem 4(c)? -

Algebra for College Students
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter13: Conic Sections
Section13.5: Systems Involving Nonlinear Equations
Problem 39PS
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89.9138336 9.4822905 11 V- Proj₁(W || = 9.4822 HW8 4(c) result
Transcribed Image Text:89.9138336 9.4822905 11 V- Proj₁(W || = 9.4822 HW8 4(c) result
3. Let (*) be the following system of five linear equations in three unknowns x₁, x2, x3 over R: 2x1 + 3x1 + (b) Suppose (b₁,b2, b3, b4, b5) x₁ + 4x₁ + -9x1 - 3x2 - 5x3 (a) Show that the system (*) has a solution if and only if b3 = 2b₁ b₂ = -2b4 - b5. x₂ + x₂ + = x3 = b₁ x3 = b₂ x₂ + x3 = = b3 x₂ + 2x3 = b4 b5. equation of the system (*), (-1, 4, 2, 4, 3). Find the corresponding normal and use it to find all least squares solutions of (*). You may use a calculator for this part. = (*) (c) Again suppose (b₁,b2, b3, b4, b5) = (-1, 4, 2, 4, 3). Instead of normal equations, now use orthogonal projection to find all least squares solutions s of (*) and the error E = ||b - As||2. Note that the necessary computation has been done in Homework 8 Problem 4(c), so you should use the result there and do this part by hand. What is the geometric meaning of the error E in the context of Homework 8 Problem 4(c)?
Transcribed Image Text:3. Let (*) be the following system of five linear equations in three unknowns x₁, x2, x3 over R: 2x1 + 3x1 + (b) Suppose (b₁,b2, b3, b4, b5) x₁ + 4x₁ + -9x1 - 3x2 - 5x3 (a) Show that the system (*) has a solution if and only if b3 = 2b₁ b₂ = -2b4 - b5. x₂ + x₂ + = x3 = b₁ x3 = b₂ x₂ + x3 = = b3 x₂ + 2x3 = b4 b5. equation of the system (*), (-1, 4, 2, 4, 3). Find the corresponding normal and use it to find all least squares solutions of (*). You may use a calculator for this part. = (*) (c) Again suppose (b₁,b2, b3, b4, b5) = (-1, 4, 2, 4, 3). Instead of normal equations, now use orthogonal projection to find all least squares solutions s of (*) and the error E = ||b - As||2. Note that the necessary computation has been done in Homework 8 Problem 4(c), so you should use the result there and do this part by hand. What is the geometric meaning of the error E in the context of Homework 8 Problem 4(c)?
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