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Linear Algebra: 14. Let A=| 1 2 | and b=| 1 |. (1) Use the Existence and Uniqueness Theorem to show Ax = b is an inconsistent linear system. (2) Find a least-squares solution to the inconsistent system Ax = b. 14. Let A=| 1 2 | and b=| 1 |. (1) Use the Existence and Uniqueness Theorem to show Ax = b is an inconsistent linear system. (2) Find a least-squares solution to the inconsistent system Ax = b.
2 9 11 and b (1 point) Let A -6 The QR factorization of the matrix A is given by: 2 1 6 17 äv2 3 1 1 0 3 2 3 21 V2 3 áva (a) Applying the QR factorization to solving the least squares problem Ax b gives the system: X = 0 2 3 (b) Use backsubstitution to solve the system in part (a) and find the least squares solution. x=
b. - 2 -1 1 and b Let A = Show that the equation Ax =b does not have a solution for all possible b, and -3 0 3 4-2 2 b3 describe the set of all b for which Ax b does have a solution How can it be shown that the equation Ax = b does not have a solution for all possible b? Choose the correct answer below. O A. Find a vector b for which the...
2.1 Suppose that AX=b does not have a solution. Such inconsistent systems often arise in applications, sometimes with large coefficient matrices. The best approximate solution is called the least-squares solution. It states that if the columns of A are linearly dependent, and the matrix AA" is invertible, the least squares solution X is given by 1 X = - ATA [ 40] 2 Find a least-squares solution of AX=b by using above expression for A = 0 2 and b...
== 2 1 3 (1 point) Let A 1 and b -3 2 6 The QR factorization of the matrix A is given by: 1 2 = ſ v2 ŠV2 0 V2 3 2 3 (a) Applying the QR factorization to solving the least squares problem Ax b gives the system: 3 wls, wie X = (b) Use backsubstitution to system in part (a) and find the least squares solution. =
110 In Problem 5.49, a least squares inverse was found for the matrix A = ſi -1 -2 -2 4 3 [ 1 1 -3 17 -2. 11 (a) Use this least squares inverse to show that the system of equations Az = C is inconsistent, where c' = (2,1,5). (b) Find a least squares solution. (c) Compute the sum of squared errors for a least squares solution to this system of equations.
a. Every matrix equation Ax b corresponds to a vector equation with the same solution set. Choose the correct answer below. O A. False. The matrix equation Ax-b does not correspond to a vector equation with the same solution set. O B. False. The matrix equation Ax b only corresponds to an inconsistent system of vector equations. O c. True. The matrix equation Ax-bis simply another notation for the vector equation x1a1 + x2a2 +·.. + xnan-b, where al ,...
Problem 6 (18 pts.): Let A be a 4 x 2 matrix given by: -1 -5 1 1 1 A= -1 -1 1 5 a) Compute the Gram-Schmidt QR factorization of A. b) Use the QR factorization to find the least squares solution of Az = 6, where 6= (-2,-1,5,0).
Suppose that А is an mxn matrix with independent columns and the equation Añ = is inconsistent. Then the following statements are true. A. The least squares solution to Ax = 6 is given by î = (A” A) A 5 B. We can reduce the least squares solution Î = (A” A)-'A” as follows. î = (A” A)'AT = Â = A-'(AT)-'A" 6 = This calculation follows since when matrices A = QR where Q = (ū ūk) and...
1-4 - 31 Let A= 3 and b= Show that the equation Ax=b does not have a solution for all possible b, and describe the set 4 26 of all b for which Ax=b does have a solution. How can it be shown that the equation Ax = b does not have a solution for all possible b? Choose the correct answer below. O A. Row reduce the augmented matrix [ a b ] to demonstrate thatſ A b )...