Homework Answers
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 syst...
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.
(4.2) Let 4 7 A= 4 7 -2 1 (a) Find the QR decomposition of A....
(4.2) Let 4 7 A= 4 7 -2 1 (a) Find the QR decomposition of A. It has to be of the form A QR where Q is a 3 x 3 orthogonal matrix, and R is 3 x 2 upper-triangular. (b) Use part (a) to find the least squares solution to the -6 Ax -4 -2
Let Ax=b be a matrix equation where A is given by 11 -2 3 2 1...
Let Ax=b be a matrix equation where A is given by 11 -2 3 2 1 and b by 3 1 2 1) Show that the equation is inconsistent. (15 points) 2) Find the least squares solution x"hat" of the equation and compute the error. (15 points)
Problem #4: Let 0 2 A = 2 9 and b = 0 2 4 Find...
Problem #4: Let 0 2 A = 2 9 and b = 0 2 4 Find the least squares solution of the linear system Ax = b. Enter the components of the least squares solution x = [x y]? into the answer box below (in order), separated with a comma. Problem #4: Enter your answer symbolically, as in these examples Just Save Submit Problem #4 for Grading Problem #4 Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark:
4. Let (Q1.Q2.Qs)T be the least squares solution of A(Q1,.Q2.Qs)T b, where r 3 -1 5 1 -1 -13 3 13...
4. Let (Q1.Q2.Qs)T be the least squares solution of A(Q1,.Q2.Qs)T b, where r 3 -1 5 1 -1 -13 3 13 7 -5 -20 -10 13 3 L 16 13 -13J Let Q - In(3+ IQil+2 Q2l+3 Qsl). Then T-5sin (100Q) satisfies: -(A) 0ST<1.
4. Let (Q1.Q2.Qs)T be the least squares solution of A(Q1,.Q2.Qs)T b, where r 3 -1 5 1 -1 -13 3 13 7 -5 -20 -10 13 3 L 16 13 -13J Let Q - In(3+ IQil+2...
7. Let 7 = (1,-1,-2), ū = (2,-1,1) and = (2,-2,-4). Find: (a) *(-20) (4 pts)...
7. Let 7 = (1,-1,-2), ū = (2,-1,1) and = (2,-2,-4). Find: (a) *(-20) (4 pts) (b) (+37). ū (4 pts) (c) The vector of magnitude 5 that points in the same direction as (4 pts). (d) The angle between 7 and ū (4 pts). (e) Find Projz() (4 pts).
1. Let {ü, 7,w, i}, where u = (3,-2), v = (0,4), ū = (-1,5) and...
1. Let {ü, 7,w, i}, where u = (3,-2), v = (0,4), ū = (-1,5) and i = (-6,4). Find the components of the resultants obtained by doing the following linear combinations. a. r = 2ū - 40 b. š= 3ū – +20 +
7. (20 points) Let 0-1 5 3 A -2 34 2 -3-5 (a) ( 15 points) Solve the linear system Ax = b by Gaus...
7. (20 points) Let 0-1 5 3 A -2 34 2 -3-5 (a) ( 15 points) Solve the linear system Ax = b by Gaussian elimination and express the general solution in vector form. (b) (5 points) Write down the corresponding homogenous system Ax-0 explicitly and determine all non-trivial solutions from (a) without resolving the system
7. (20 points) Let 0-1 5 3 A -2 34 2 -3-5 (a) ( 15 points) Solve the linear system Ax = b by...
== 2 1 3 (1 point) Let A 1 and b -3 2 6 The QR...
== 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. =
eclass.srv.ualberta.ca 2 of 2 1. Consider the matrix 3-2 1 4-1 2 3 5 7 8...
eclass.srv.ualberta.ca 2 of 2 1. Consider the matrix 3-2 1 4-1 2 3 5 7 8 (a) Find a basis B for the null space of A. Hint: you need to verify that the vectors you propose 20 actually form a basis for the null space. (Recall: (1) the null space of A consists of all x e R with Ax = 0, and (2) the matrix equation Ax = 0 is equivalent to a certain system of linear equations.)...
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.
(4.2) Let 4 7 A= 4 7 -2 1 (a) Find the QR decomposition of A. It has to be of the form A QR where Q is a 3 x 3 orthogonal matrix, and R is 3 x 2 upper-triangular. (b) Use part (a) to find the least squares solution to the -6 Ax -4 -2
Let Ax=b be a matrix equation where A is given by 11 -2 3 2 1 and b by 3 1 2 1) Show that the equation is inconsistent. (15 points) 2) Find the least squares solution x"hat" of the equation and compute the error. (15 points)
Problem #4: Let 0 2 A = 2 9 and b = 0 2 4 Find the least squares solution of the linear system Ax = b. Enter the components of the least squares solution x = [x y]? into the answer box below (in order), separated with a comma. Problem #4: Enter your answer symbolically, as in these examples Just Save Submit Problem #4 for Grading Problem #4 Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark:
4. Let (Q1.Q2.Qs)T be the least squares solution of A(Q1,.Q2.Qs)T b, where r 3 -1 5 1 -1 -13 3 13 7 -5 -20 -10 13 3 L 16 13 -13J Let Q - In(3+ IQil+2 Q2l+3 Qsl). Then T-5sin (100Q) satisfies: -(A) 0ST<1.
4. Let (Q1.Q2.Qs)T be the least squares solution of A(Q1,.Q2.Qs)T b, where r 3 -1 5 1 -1 -13 3 13 7 -5 -20 -10 13 3 L 16 13 -13J Let Q - In(3+ IQil+2...
7. Let 7 = (1,-1,-2), ū = (2,-1,1) and = (2,-2,-4). Find: (a) *(-20) (4 pts) (b) (+37). ū (4 pts) (c) The vector of magnitude 5 that points in the same direction as (4 pts). (d) The angle between 7 and ū (4 pts). (e) Find Projz() (4 pts).
1. Let {ü, 7,w, i}, where u = (3,-2), v = (0,4), ū = (-1,5) and i = (-6,4). Find the components of the resultants obtained by doing the following linear combinations. a. r = 2ū - 40 b. š= 3ū – +20 +
7. (20 points) Let 0-1 5 3 A -2 34 2 -3-5 (a) ( 15 points) Solve the linear system Ax = b by Gaussian elimination and express the general solution in vector form. (b) (5 points) Write down the corresponding homogenous system Ax-0 explicitly and determine all non-trivial solutions from (a) without resolving the system
7. (20 points) Let 0-1 5 3 A -2 34 2 -3-5 (a) ( 15 points) Solve the linear system Ax = b by...
== 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. =
eclass.srv.ualberta.ca 2 of 2 1. Consider the matrix 3-2 1 4-1 2 3 5 7 8 (a) Find a basis B for the null space of A. Hint: you need to verify that the vectors you propose 20 actually form a basis for the null space. (Recall: (1) the null space of A consists of all x e R with Ax = 0, and (2) the matrix equation Ax = 0 is equivalent to a certain system of linear equations.)...
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