- In Exercises 1 4 Determine If The System Has A Nontrivial Solution Try To Use As Few Row Operations As Possible 1 2 1 (471.45 KiB) Viewed 91 times
In Exercises 1-4, determine if the system has a nontrivial solution. Try to use as few row operations as possible. 1. 2. 2xı - 5x2 + 8x3 = 0 -2xı - 7x2 + x3 = 0 4xı + 2x2 + 7x3 = 0 x] - 3x2 + 7x3 = 0 -2x1 + x2 - 4x3 = 0 x1 + 2x2 + 9x3 = 0 3. -3x1 + 5x2 - 7x3 = 0 4.-5x + 7x2 + 9x3 = 0 -6x1 + 7x2 + x3 = 0 x1 - 2x2 + 6x3 = 0 In Exercises 5 and 6, follow the method of Examples 1 and 2 to write the solution set of the given homogeneous system in parametric vector form. 13. Verify that the solutions you found to Exercise 9 are indeed homogeneous solutions. 14. Verify that the solutions you found to Exercise 10 are indeed homogeneous solutions. 15. Verify that the solutions you found to Exercise 11 are indeed homogeneous solutions. 16. Verify that the solutions you found to Exercise 12 are indeed homogeneous solutions. 17. Suppose the solution set of a certain system of linear equa- tions can be described as xi = 5 + 4x3, x2 = -2 - 7x3, with x3 free. Use vectors to describe this set as a line in R. 18. Suppose the solution set of a certain system of linear equations can be described as x = 3x4, x2 = 8 + x4, x3 = 2 - 5x4, with x4 free. Use vectors to describe this set as a line in R4 5. x1 + 3x2 + x3 = 0 6. x + 3x2 - 5x3 = 0 - 4x1 - 9x2 + 2x3 = 0 X+ 4x2 - 8x3 = 0 - 3x2 - 6x3 = 0 -3x1 - 7x2 + 9x3 = 0 In Exercises 7-12, describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix. 8. 1 0 -2 -9 5 1 2 -6 7- [5 =] » [-3_, 19. Follow the method of Example 3 to describe the solutions of the following system in parametric vector form. Also, give a geometric description of the solution set and compare it to that in Exercise 5. X1 + 3x2 + x3 = 1 - 4x1 - 9x2 + 2x3 = -1 - 3x2 - 6x3 = -3 10. 1 2 3 6 0 - 4 0 - 8