Solve the n equations for the n variables x1,x2, . . . , rn respectively. 1-12, .
Example 97 We can find general formulas that characterize the procedure used in the previous example. Suppose we want to solve the equation Ux = v, where x = (x)' and v-(v)' are column vectors. Then we may write the system of equations corresponding to this matrix equation as: n. We may solve these n equations for the n variables 1, 2,... . Xn respectively: 1 1 n. n. Now, beginning at the last equation and working back to the first, we may solve for the variables z',z"-1, . . .12.ri one by one, as long as u; 0 for all J-1, . . . ,n. There is a similar formula for the solution of Lx = v, which we leave as an exercise
Homework Answers
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(0) is a lower- Consider the matrix equation Lx u, where L triangular square matrix and x = (p" 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 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...
4. Chemical Reaction: Consider the following chemical equation: Where x , y , z ,u and...
4. Chemical Reaction: Consider the following chemical equation: Where x , y , z ,u and v are the numbers of molecules of NH3 , , H, , H,0 and . respectively, to be determined. [Here, a molecule with the symbol NH indicates that it contains 1 atom of the element N and 3 atoms of the element H. Similar interpretation applies to the other molecules in the equation.] Given that the total number of atoms for each of the...
3. Using separation of variables to solve the heat equation, u- kuxx on the interval 0 < x< 1 with boundary conditions ux(0, t) = 0 and ux(1, t) yields the general solution, 1, 0<x < 1/2...
3. Using separation of variables to solve the heat equation, u- kuxx on the interval 0 < x< 1 with boundary conditions ux(0, t) = 0 and ux(1, t) yields the general solution, 1, 0<x < 1/2 0, 1/2 x<1 Determine the coefficients An (n = 0, 1, 2, . . .) when u(x,0) = f(x) =
3. Using separation of variables to solve the heat equation, u- kuxx on the interval 0
Theorem. Consider the quadratic form Q(x) = Ar where A is anxn symmetric matrix and A,...
Theorem. Consider the quadratic form Q(x) = Ar where A is anxn symmetric matrix and A, and denote the largest and smallest eigenvalues of A, respectively. Then max Q(x) = 2 = max Q() = 1 and Q0.) = 1, where is any unit eige vector corre sponding to ii) in (r) and QU.) where is any unit eigen vector corresponding to do 1. - Find max Q(x) and min Q(x). 1) Q(1) = 3x + 43273 +673 ii) Q(z)...
Consider a DTMC X;n 2 0 with state space E 0,1,2,... ,N), and transition probability matrix P = (...
Consider a DTMC X;n 2 0 with state space E 0,1,2,... ,N), and transition probability matrix P = (pij). Define T = min(n > 0 : Xn-0), and vi(n) = P(T > n|X0 = i). Use the first-step analysis to show that vi (72), t"2(n), . . . , UN(n)) = where B is a submatrix of P obtained by deleting the row and column corresponding to the state 0. Hint: First establish a recursive formula v(n )-ΣΝ1pijuj(n-1).
Consider a...
Let A be an m × n matrix, let x Rn and let 0 be the zero vector in Rm. (a) Let u, v є Rn be any two solutions of Ax 0,...
Let A be an m × n matrix, let x Rn and let 0 be the zero vector in Rm. (a) Let u, v є Rn be any two solutions of Ax 0, and let c E R. Use the properties of matrix-vector multiplication to show that u+v and cu are also solutions of Ax O. (b) Extend the result of (a) to show that the linear combination cu + dv is a solution of Ax 0 for any c,d...
differential equation 5. Solve the system, where x and x' represent column vectors as discussed in class. J...
differential equation
5. Solve the system, where x and x' represent column vectors as discussed in class. J where n is your computer number (if you were not in class when the Exam was handed out, let n be the score on your Exam 1.) For what values of ? will x(0) this system be stable? Then solve the system with result in complete sentences. Show all work. 」and interpret your We were unable to transcribe this image
5. Solve...
(1 point) The equation 3ry2r 2y2 (*) can be written in the form y f(y/x), ie.,...
(1 point) The equation 3ry2r 2y2 (*) can be written in the form y f(y/x), ie., it is homogeneous, so we can use the substitution u = y/x to obtain a separable equation with dependent variable uu(x. Introducing this substitution and using the fact that y' ru' u we can write () as y xu'w = f(u) where f(u) Separating variables we can write the equation in the form da np (n)6 where g(u) = An implicit general solution with...
Use the Fourier transform to find a solution of the ordinary differential equation u´´-u+2g(x) =0 where g∈L1. (The solu...
Use the Fourier transform to find a solution of the ordinary
differential equation u´´-u+2g(x) =0
where g∈L1. (The solution obtained this
way is the one that vanishes at ±∞. What is the general
solution?)
1. Use the Fourier transform to find a solution of the ordinary differential equation u" - u + 2g(x) = 0 where g E L. (The solution obtained this way is the one that vanishes at £oo. What is the general solution?) eg(y)dy eg(y)dy e Answer:...
where V is an n × n orthogonal matrix and U is an m × m orthogonal matrix with entries σί, , , , , Ơr where r min{m, n), one can show that A 3 Computation of an SVD We will now compute the SVD o...
where V is an n × n orthogonal matrix and U is an m × m orthogonal matrix with entries σί, , , , , Ơr where r min{m, n), one can show that A 3 Computation of an SVD We will now compute the SVD of a simple 3 × 2 matrix. Let Answer the following questions to compute the SVD of A. 5, Determine a bases for the eigenspace of λ-11and λ-1. 6. Lastly normalize the vectors (mske...
(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...
4. Chemical Reaction: Consider the following chemical equation: Where x , y , z ,u and v are the numbers of molecules of NH3 , , H, , H,0 and . respectively, to be determined. [Here, a molecule with the symbol NH indicates that it contains 1 atom of the element N and 3 atoms of the element H. Similar interpretation applies to the other molecules in the equation.] Given that the total number of atoms for each of the...
3. Using separation of variables to solve the heat equation, u- kuxx on the interval 0 < x< 1 with boundary conditions ux(0, t) = 0 and ux(1, t) yields the general solution, 1, 0<x < 1/2 0, 1/2 x<1 Determine the coefficients An (n = 0, 1, 2, . . .) when u(x,0) = f(x) =
3. Using separation of variables to solve the heat equation, u- kuxx on the interval 0
Theorem. Consider the quadratic form Q(x) = Ar where A is anxn symmetric matrix and A, and denote the largest and smallest eigenvalues of A, respectively. Then max Q(x) = 2 = max Q() = 1 and Q0.) = 1, where is any unit eige vector corre sponding to ii) in (r) and QU.) where is any unit eigen vector corresponding to do 1. - Find max Q(x) and min Q(x). 1) Q(1) = 3x + 43273 +673 ii) Q(z)...
Consider a DTMC X;n 2 0 with state space E 0,1,2,... ,N), and transition probability matrix P = (pij). Define T = min(n > 0 : Xn-0), and vi(n) = P(T > n|X0 = i). Use the first-step analysis to show that vi (72), t"2(n), . . . , UN(n)) = where B is a submatrix of P obtained by deleting the row and column corresponding to the state 0. Hint: First establish a recursive formula v(n )-ΣΝ1pijuj(n-1).
Consider a...
Let A be an m × n matrix, let x Rn and let 0 be the zero vector in Rm. (a) Let u, v є Rn be any two solutions of Ax 0, and let c E R. Use the properties of matrix-vector multiplication to show that u+v and cu are also solutions of Ax O. (b) Extend the result of (a) to show that the linear combination cu + dv is a solution of Ax 0 for any c,d...
differential equation
5. Solve the system, where x and x' represent column vectors as discussed in class. J where n is your computer number (if you were not in class when the Exam was handed out, let n be the score on your Exam 1.) For what values of ? will x(0) this system be stable? Then solve the system with result in complete sentences. Show all work. 」and interpret your We were unable to transcribe this image
5. Solve...
(1 point) The equation 3ry2r 2y2 (*) can be written in the form y f(y/x), ie., it is homogeneous, so we can use the substitution u = y/x to obtain a separable equation with dependent variable uu(x. Introducing this substitution and using the fact that y' ru' u we can write () as y xu'w = f(u) where f(u) Separating variables we can write the equation in the form da np (n)6 where g(u) = An implicit general solution with...
Use the Fourier transform to find a solution of the ordinary
differential equation u´´-u+2g(x) =0
where g∈L1. (The solution obtained this
way is the one that vanishes at ±∞. What is the general
solution?)
1. Use the Fourier transform to find a solution of the ordinary differential equation u" - u + 2g(x) = 0 where g E L. (The solution obtained this way is the one that vanishes at £oo. What is the general solution?) eg(y)dy eg(y)dy e Answer:...
where V is an n × n orthogonal matrix and U is an m × m orthogonal matrix with entries σί, , , , , Ơr where r min{m, n), one can show that A 3 Computation of an SVD We will now compute the SVD of a simple 3 × 2 matrix. Let Answer the following questions to compute the SVD of A. 5, Determine a bases for the eigenspace of λ-11and λ-1. 6. Lastly normalize the vectors (mske...
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