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4. (30 points) Consider the following 2 x 2 system Axb (a) (10 points) Use the...
(a) Suppose we want to solve the linear vector-matrix equation Ax b for the vector x. Show that the Gauss elimination algorithm may be written bAbm,B where m 1, This process produces a matrix equa...
(a) Suppose we want to solve the linear vector-matrix equation Ax b for the vector x. Show that the Gauss elimination algorithm may be written bAbm,B where m 1, This process produces a matrix equation of the form Ux = g , in which matrix U is an upper-triangular matrix. Show that the solution vector x may be obtained by a back-substitution algorithm, in the form Jekel (b) Iterative methods for solving Ax-b work by splitting matrix A into two...
2. (a) Let A be the matrix A -4 21 8 -40 Write down the 3 x 3 permutation matrix P such that PA interchanges the 1st and 3rd rows of A. Find the inverse of P. Use Gaussian elimination with partial pi...
2. (a) Let A be the matrix A -4 21 8 -40 Write down the 3 x 3 permutation matrix P such that PA interchanges the 1st and 3rd rows of A. Find the inverse of P. Use Gaussian elimination with partial pivoting to find an upper triangular matrix U, permutation matrices Pi and P2 and lower triangular matrices Mi and M2 of the form 1 0 0 Mi-1A1 10 a2 0 1 M2 0 0 0 b1 with ail...
ALTSIS AND NUMERICAL ANALYSIS 2. (a) Let A be the matrix 2 -115 8-4 Write down the 3 x 3 permutat...
ALTSIS AND NUMERICAL ANALYSIS 2. (a) Let A be the matrix 2 -115 8-4 Write down the 3 x 3 permutation matrix P such that PA interchanges the 1st and 3rd rows of A. Find the inverse of P Use Gaussian elimination with partial pivoting to find an upper triangular matix U, permutation matrices Pi and P2 and lower triangular matrices M and M2 of the form 1 0 0 0 1 1 0 0 0 bi 1 with land...
Solve the system Ux = y for x. U = ? X = ? If the...
Solve the system Ux =
y for x.
U = ?
X = ?
If the nxn matrix A can be expressed as A = LU, where L is a lower triangular matrix and U is an upper triangular matrix, then the system Ax = b can be expressed as LUX = b and can be solved in two steps: Step 1. Let Ux = y, so that LUX = b can be expressed as Ly = b. Solve this...
Problem 13.13. Consider the system of three linear differential equations: xt = 2x1 + 3x2 +...
Problem 13.13. Consider the system of three linear differential equations: xt = 2x1 + 3x2 + 4.13 where the unknowns are the three functions xi(t), x2(t), and 23(t). x'a = 2x2 + 6.13 (a) Write the system in the form x' = Ax, where A is a (3 x 3) matrix. X'z = 2x3 (b) Write A as the sum of two matrices, A=D+U, where D is a diagonal matrix (all of the off-diagonal entries are zero, and the diagonal...
HW10P5 (10 points) 3 2 -1 Let A be the matrix A = 1-3 0 6...
HW10P5 (10 points) 3 2 -1 Let A be the matrix A = 1-3 0 6 -2 1 a. (4 pts) Find the multipliers l21, 131,132 and the elemention matrices E21, E31, E32 b. (2 pts) Use the multipliers l21, 131,132 to construct the lower triangular matrix, L c. (2 pts) Use the elimination matrices to determine the upper triangular, U, matrix of A d. (2 pts) verify that LU A
linear algebra Let V (71, 72, 3}, where 71 73=(2,0,3). (1,3,-1), 2 = (0, 1,4), and...
linear algebra
Let V (71, 72, 3}, where 71 73=(2,0,3). (1,3,-1), 2 = (0, 1,4), and (a) Prove: V is a basis. (b) Find the coordinates of (b, b2, bs) with respect to V = {71, U2, 3,}. (c) Suppose M and M' are matrices whose columns span the same vector space V. Let b be the coordinates of relative to M. Write a matrix equation that gives b', the coordinates of relative to M'. (Your answer should be a...
Please help me for all problems 1, 2, 3, 4, 5 1. (Three points.) Convert this...
Please help me for all problems 1, 2, 3, 4, 5
1. (Three points.) Convert this system to upper triangular form and solve by back-substitution. 4x+7y + 5z 13 -2y + 2z-6 2. (Three points.) Convert this system to upper triangular form and solve by back-substitution. 4x-5y +z=-13 2x -y-3z5 3. (Four points.) Find the value a that will make the matrix of coefficients for this system singular and the value b that will give the system infinitely many solutions...
4. Consider the following matrix [1 0 -27 A=000 L-2 0 4] (a) (3 points) Find...
4. Consider the following matrix [1 0 -27 A=000 L-2 0 4] (a) (3 points) Find the characteristic polynomial of A. (b) (4 points) Find the eigenvalues of A. Give the algebraic multiplicity of each eigenvalue (c) (8 points) Find the eigenvectors corresponding to the eigenvalues found in part (b). (d) (4 points) Give a diagonal matrix D and an invertible matrix P such that A = PDP-1 (e) (6 points) Compute P-and verify that A= PDP- (show your steps).
1. [A] is the coefficient matrix for [Aj[X]-(C. 12-10 16 A-16 9 24 12 8 At the end of forward elimination steps of Gaussian Elimination method with partial pivoting, the coefficient matrix looks...
1. [A] is the coefficient matrix for [Aj[X]-(C. 12-10 16 A-16 9 24 12 8 At the end of forward elimination steps of Gaussian Elimination method with partial pivoting, the coefficient matrix looks like 0 0 by a) bs is most nearly (circle correct response) [10 pts.] A. -2.0298 B. 1.4167 C. 12.000 D. 22.667 b) This is a consistent/inconsistent system. (circle correct response) (5 points) A square matrix [A] is upper triangular if (circle correct response) |5 points (A)...
(a) Suppose we want to solve the linear vector-matrix equation Ax b for the vector x. Show that the Gauss elimination algorithm may be written bAbm,B where m 1, This process produces a matrix equation of the form Ux = g , in which matrix U is an upper-triangular matrix. Show that the solution vector x may be obtained by a back-substitution algorithm, in the form Jekel (b) Iterative methods for solving Ax-b work by splitting matrix A into two...
2. (a) Let A be the matrix A -4 21 8 -40 Write down the 3 x 3 permutation matrix P such that PA interchanges the 1st and 3rd rows of A. Find the inverse of P. Use Gaussian elimination with partial pivoting to find an upper triangular matrix U, permutation matrices Pi and P2 and lower triangular matrices Mi and M2 of the form 1 0 0 Mi-1A1 10 a2 0 1 M2 0 0 0 b1 with ail...
ALTSIS AND NUMERICAL ANALYSIS 2. (a) Let A be the matrix 2 -115 8-4 Write down the 3 x 3 permutation matrix P such that PA interchanges the 1st and 3rd rows of A. Find the inverse of P Use Gaussian elimination with partial pivoting to find an upper triangular matix U, permutation matrices Pi and P2 and lower triangular matrices M and M2 of the form 1 0 0 0 1 1 0 0 0 bi 1 with land...
Solve the system Ux =
y for x.
U = ?
X = ?
If the nxn matrix A can be expressed as A = LU, where L is a lower triangular matrix and U is an upper triangular matrix, then the system Ax = b can be expressed as LUX = b and can be solved in two steps: Step 1. Let Ux = y, so that LUX = b can be expressed as Ly = b. Solve this...
Problem 13.13. Consider the system of three linear differential equations: xt = 2x1 + 3x2 + 4.13 where the unknowns are the three functions xi(t), x2(t), and 23(t). x'a = 2x2 + 6.13 (a) Write the system in the form x' = Ax, where A is a (3 x 3) matrix. X'z = 2x3 (b) Write A as the sum of two matrices, A=D+U, where D is a diagonal matrix (all of the off-diagonal entries are zero, and the diagonal...
HW10P5 (10 points) 3 2 -1 Let A be the matrix A = 1-3 0 6 -2 1 a. (4 pts) Find the multipliers l21, 131,132 and the elemention matrices E21, E31, E32 b. (2 pts) Use the multipliers l21, 131,132 to construct the lower triangular matrix, L c. (2 pts) Use the elimination matrices to determine the upper triangular, U, matrix of A d. (2 pts) verify that LU A
linear algebra
Let V (71, 72, 3}, where 71 73=(2,0,3). (1,3,-1), 2 = (0, 1,4), and (a) Prove: V is a basis. (b) Find the coordinates of (b, b2, bs) with respect to V = {71, U2, 3,}. (c) Suppose M and M' are matrices whose columns span the same vector space V. Let b be the coordinates of relative to M. Write a matrix equation that gives b', the coordinates of relative to M'. (Your answer should be a...
Please help me for all problems 1, 2, 3, 4, 5
1. (Three points.) Convert this system to upper triangular form and solve by back-substitution. 4x+7y + 5z 13 -2y + 2z-6 2. (Three points.) Convert this system to upper triangular form and solve by back-substitution. 4x-5y +z=-13 2x -y-3z5 3. (Four points.) Find the value a that will make the matrix of coefficients for this system singular and the value b that will give the system infinitely many solutions...
4. Consider the following matrix [1 0 -27 A=000 L-2 0 4] (a) (3 points) Find the characteristic polynomial of A. (b) (4 points) Find the eigenvalues of A. Give the algebraic multiplicity of each eigenvalue (c) (8 points) Find the eigenvectors corresponding to the eigenvalues found in part (b). (d) (4 points) Give a diagonal matrix D and an invertible matrix P such that A = PDP-1 (e) (6 points) Compute P-and verify that A= PDP- (show your steps).
1. [A] is the coefficient matrix for [Aj[X]-(C. 12-10 16 A-16 9 24 12 8 At the end of forward elimination steps of Gaussian Elimination method with partial pivoting, the coefficient matrix looks like 0 0 by a) bs is most nearly (circle correct response) [10 pts.] A. -2.0298 B. 1.4167 C. 12.000 D. 22.667 b) This is a consistent/inconsistent system. (circle correct response) (5 points) A square matrix [A] is upper triangular if (circle correct response) |5 points (A)...
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