please help !!!! 10. 20 points Consider the homogeneous system x' Ax, where 4 0 0 A 1 0 2 02 3 a) Show that v = | 1 | and w = 1-2) are eigenvectors of A. b) Identify the defective eigenvalu...
I know A-D. Please do E-G only. Thanks! [ 1 ] [ 0 ] = W, W_2 is found in part F [ 1 ] 3. (Taken from Boyce & DiPrima) Consider the 3-dimensional system of linear equations Ti 11] X' = AX = 2 1 -1 x 1-3 2 4 (a) Show that the three eigenvalues of the coefficient matrix, A, are 1, = lyd = 2. This is an eigenvalue of multiplicity 3. (b) Show that all the...
4. (15 pts Consider the following direction fields IV VI (5 pts)Which of the direction fields corresponds to the system x -Ax, where A is a 2x2 matrix with eigenvalues λ,--1 and λ2-2 and corresponding eigenvectors vand v- 1? a. is a 2x2 matrix with repeated eigenvalue λ = 0 with defect 1 (has only one linearly independent eigenvector, not two.) and corresponding eigenvector vi- 13 (5 pts) Which of the direction fields corresponds to the system x -Cx, where...
4. (3 points each) Consider the system y' = Cy, where C = a) Find the eigenvalues of C. (b) Find the eigenvectors corresponding to each eigenvalue of C. (c) Write down a general solution to the system.
7. (10 points) Find the general solution to the homogeneous system of DE: x' = Ax where A = [-2 21
I need some help on this problem. Thank you! 15. Consider the differential equation (*) х%3D Ах + vex where v is an eigenvector of A with eigenvalue X. Suppose moreover, that A has linearly independent eigenvectors v', v2,... , v", with distinct eigenvalues Ар А, гespectively. (a) Show that (*) has no solution /(t) of the form V(t)= ae*. Hint: Write a= а у+.. +a, v. (b) Show that (*) has a solution /(t) of the form n 0)...
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...
Question B3 (10 marks) Solve the following homogeneous system of first order ODE dai di da dt x,(0)=2. Makesures ou usethe initial with the initial conditions (0):0 0)=1, is in the following form Hint: It is given that one of the solutions of the above system Useful formulas Case 1: A only have distinct roots for λ General solution is ,n) are the coresponding eigenvectors where K, (where i = 1,2, Case 2: For a system of n equations, the...
2. (10 points) Suppose v is an eigenvector of A with eigenvalue X, and let c be a real number. Show that v is an eigenvector of A+cI, where I is the appropriately sized identity matrix. What is the corresponding eigenvalue?
7. (10 points) Find the general solution to the homogeneous system of DE: -1 x' = Ax where A -2 = [ 21
2 8. (10 points) Find the general solution to the homogeneous system of DE: | 3 x' = Ax where A = -2 1 - 1 -2 -4.