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)? -
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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Transcribed Image Text:89.9138336
9.4822905
11 V- Proj₁(W || = 9.4822
HW8 4(c)
result
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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