Homework Answers
3 and 4 The matrix equation (Ax b) -1 -2 -1 1 2 2 0 1 has no solution. We wish to find the best approximate solution to...
3 and 4
The matrix equation (Ax b) -1 -2 -1 1 2 2 0 1 has no solution. We wish to find the best approximate solution to this system 1. Write the system of equations used to find the best approximation (i.c., write the system corresponding to the "normal equations"). Preview Preview 2. The solution to the system of normal cquations is Preview 3. The vector in the column space of A nearest to the vector b is Preview...
Problem 5 (a) Let A be an n × m matrix, and suppose that there exists a m × n matrix B such that BA = 1- (i) Let b є Rn be such that the system of equations Ax b has at least one solution. Prove that...
Problem 5 (a) Let A be an n × m matrix, and suppose that there exists a m × n matrix B such that BA = 1- (i) Let b є Rn be such that the system of equations Ax b has at least one solution. Prove that this solution must be unique. (ii) Must it be the case that the system of equations Ax = b has a solution for every b? Prove or provide a counterexample. (b) Let...
4. Consider the matrix [1 0 01 A- 1 0 2-1and the vector b2 (a) Construct...
4. Consider the matrix [1 0 01 A- 1 0 2-1and the vector b2 (a) Construct the augmented matrix [Alb] and use elementary row operations to trans- form it to reduced row echelon form. (b) Find a basis for the column space of A. (c) Express the vectors v4 and vs, which are column vectors of column 4 and 5 of A, as linear combinations of the vectors in the basis found in (b) (d) Find a basis for the...
T042 Consider A-2 1 2 0 0 and B -- This overdetermined Ax-B system cannot be solved exactly. The ...
T042 Consider A-2 1 2 0 0 and B -- This overdetermined Ax-B system cannot be solved exactly. The backslash operator gives an approximate 'best' solution. Determine the residual r Ax-B, where x'is the approximation given by the backslash operator. Your answer is the 1-norm of the residual r, rounded to the nearest 0.1 decimal Answer: Check
T042 Consider A-2 1 2 0 0 and B -- This overdetermined Ax-B system cannot be solved exactly. The backslash operator gives an...
a. Every matrix equation Ax b corresponds to a vector equation with the same solution set....
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 ,...
Let A e Rmxn. The linear system Ax = b can have either: (i) a unique...
Let A e Rmxn. The linear system Ax = b can have either: (i) a unique solution, (ii) no solution, or (iii) infinitely many solutions. If A is square and invertible, there is a unique solution, which can be written as x = A-'b. The concept of pseudoinverse seeks to generalise this idea to non-square matrices and to cases (ii) and (iii). Taking case (ii) of an inconsistent linear system, we may solve the normal equations AT Ar = Ab...
Suppose A is a symmetric 3 by 3 matrix with eigenvalues 0, 1, 2 (a) What...
Suppose A is a symmetric 3 by 3 matrix with eigenvalues 0, 1, 2 (a) What properties 4. can be guaranteed for the corresponding unit eigenvectors u, v, w? In terms of u, v, w describe the nullspace, left nullspace, (b) row space, and column space of A (c) Find a vector x that satisfies Ax v +w. Is x unique? Under what conditions on b does Ax = b have a solution? (d) (e) If u, v, w are...
6) Suppose a matrix equation, Ax = b, has two solutions and ༼ཡང བ find an...
6) Suppose a matrix equation, Ax = b, has two solutions and ༼ཡང བ find an infinite number of column vector solutions parameterized by t. (Hint: try finding a solution to the homogenous equation Ax = 0.)
a) Write the augmented matrix for the linear system that corresponds to the matrix equation AX...
a) Write the augmented matrix for the linear system that
corresponds to the matrix equation AX = b
b) solve the above system
c) write the solution as a vector
A= 0 4 5 -1 3 2 and b= 1 24 5 4 2
b. - 2 -1 1 and b Let A = Show that the equation Ax =b...
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...
3 and 4
The matrix equation (Ax b) -1 -2 -1 1 2 2 0 1 has no solution. We wish to find the best approximate solution to this system 1. Write the system of equations used to find the best approximation (i.c., write the system corresponding to the "normal equations"). Preview Preview 2. The solution to the system of normal cquations is Preview 3. The vector in the column space of A nearest to the vector b is Preview...
Problem 5 (a) Let A be an n × m matrix, and suppose that there exists a m × n matrix B such that BA = 1- (i) Let b є Rn be such that the system of equations Ax b has at least one solution. Prove that this solution must be unique. (ii) Must it be the case that the system of equations Ax = b has a solution for every b? Prove or provide a counterexample. (b) Let...
4. Consider the matrix [1 0 01 A- 1 0 2-1and the vector b2 (a) Construct the augmented matrix [Alb] and use elementary row operations to trans- form it to reduced row echelon form. (b) Find a basis for the column space of A. (c) Express the vectors v4 and vs, which are column vectors of column 4 and 5 of A, as linear combinations of the vectors in the basis found in (b) (d) Find a basis for the...
T042 Consider A-2 1 2 0 0 and B -- This overdetermined Ax-B system cannot be solved exactly. The backslash operator gives an approximate 'best' solution. Determine the residual r Ax-B, where x'is the approximation given by the backslash operator. Your answer is the 1-norm of the residual r, rounded to the nearest 0.1 decimal Answer: Check
T042 Consider A-2 1 2 0 0 and B -- This overdetermined Ax-B system cannot be solved exactly. The backslash operator gives an...
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 ,...
Let A e Rmxn. The linear system Ax = b can have either: (i) a unique solution, (ii) no solution, or (iii) infinitely many solutions. If A is square and invertible, there is a unique solution, which can be written as x = A-'b. The concept of pseudoinverse seeks to generalise this idea to non-square matrices and to cases (ii) and (iii). Taking case (ii) of an inconsistent linear system, we may solve the normal equations AT Ar = Ab...
Suppose A is a symmetric 3 by 3 matrix with eigenvalues 0, 1, 2 (a) What properties 4. can be guaranteed for the corresponding unit eigenvectors u, v, w? In terms of u, v, w describe the nullspace, left nullspace, (b) row space, and column space of A (c) Find a vector x that satisfies Ax v +w. Is x unique? Under what conditions on b does Ax = b have a solution? (d) (e) If u, v, w are...
6) Suppose a matrix equation, Ax = b, has two solutions and ༼ཡང བ find an infinite number of column vector solutions parameterized by t. (Hint: try finding a solution to the homogenous equation Ax = 0.)
a) Write the augmented matrix for the linear system that
corresponds to the matrix equation AX = b
b) solve the above system
c) write the solution as a vector
A= 0 4 5 -1 3 2 and b= 1 24 5 4 2
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...
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