INSTRUCTIONS:

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1. SYSTEMS OF LINEAR EQUATIONS (8) Consider the following system of linear equations ( where b1, . . ., b are constants). u + 20 - w - 2x+ 3y = b1 E- y + 2= = 62 2u + 40 -2w - 4x + 7y -4= = b3 -at y - 22 = by 3u + 6v - 3w - 6+ + 7y +8: = b; (a) In the process of Gaussian elimination the leading variables of this system are and the free variables are (b) What condition(s) must the constants by, ...,b satisfy so that the system is consis- tent? Answer: (c) Do the numbers by = 1, by = -3. by = 2. by = by = 3 satisfy the condition(s) you listed in (b)? . If so, find the general solution to the system as a function of the free variables. Answer: V= (9) Consider the following homogeneous system of linear equations ( where a and b are nonzero constants). + + 2y =0 of + 8y + 3= = 0 by + 52 = 0 (a) Find a value for a which will make it necessary during Gaussian elimination to inter- change rows in the coefficient matrix. Answer: a = (b) Suppose that a does not have the value you found in part (a). Find a value for b so that the system has a nontrivial solution. Answer: b = 6 + go where c = _ _ and d = (c) Suppose that a does not have the value you found in part (a) and that b = 100. Suppose further that a is chosen so that the solution to the system is not unique. The general solution to the system (in terms of the free variable) is ( ? = , - }2, =) where o = and 3 =1.3. PROBLEMS 7 1.3. Problems (1) Give a geometric description of a single linear equation in three variables. Then give a geometric description of the solution set of a system of 3 linear equations in 3 variables if the system (a) is inconsistent. (b) is consistent and has no free variables. (c) is consistent and has exactly one free variable. (d) is consistent and has two free variables. (2) Consider the following system of equations: -misty = b -mar + y = b2 (# ) (a) Prove that if my # ma, then (*) has exactly one solution. What is it? (b) Suppose that my = me. Then under what conditions will (+) be consistent? (c) Restate the results of (a) and (b) in geometrical language