Here are solutions to some of the * and additional exercises for section 1.5.

Section 1.5   1,3,5-7,11-15,17-23o,25*, 27*, 29-33, 40*, plus the following:

  25.   Solution (part a):  We must show that Aw = b.  Observe that Aw = A(p+v_h) =  Ap + Av_h, by Theorem 5a.   But we also know that Ap = b and Av_h = 0. Substitution therefore gives us Aw  =  b + 0 = b.  That was what we wished to show.

Solution (part b):  We must show that Av_h = 0.  Since w = p+v_h ,   we can also write v_h = w - p.  Therefore we have

           Av_h =  A(w - p) = Aw - Ap = b - b = 0,

where the second equality is justfied by Theorem 5a.  This shows that Av_h = 0, as required.

101. Define homogeneous system, trivial solution, nontrivial solution, parametric vector equation
102. (*) Three students are independently working on solving an inhomogeneous linear system of the form A x = b. The first student finds a solution vector x. The second student finds a different solution vector y. Did someone make an error?
     Not necessarily.  If the system is consistent and has more than one solution, then different students might easily have found different correct answers.
     A third student finds yet another solution, z. The three students notice that x + y = z. Did someone make an error?
     Yes.  Since the system is described as inhomogeneous, we may assume that b is not zero.  In that case, the following equations cannot all be true:
     Ax = b,  Ay = b,  Az = b,  x + y = z.  In particular, if z = x + y, then x + y - z = 0.  Now apply  A to both sides, noting that A0 = 0:
                                                   0 = A0 = A(x + y - z) = Ax + Ay - Az = b + b - b = b.  
     Since this implies that b = 0, which we know is false, at least one of the four equations we started with must be incorrect. 
     
     
103. (*) Prove: If a and b are solutions to a homogeneous system, then so is a + b.
     answer: A homogeneous system has the form Ax = 0.  So if a and b are both solutions to this system that means that Aa = 0 = Ab.  Therefore, A(a + b) = Aa + Ab = 0 + 0 = 0.  This shows that a + b is a solution to the same homogeneous solution.
     
104. Prove: If a is a solution to a homogeneous system, then so is ra for any real number r.
105. (*) Prove: If a and b are solutions to a homogeneous system, then so is ra+ sb, for any real numbers r and s.
     answer: A homogeneous system has the form Ax = 0.  So if a and b are both solutions to this system, and if r and s are real numbers, we have  A(ra + sb) = A(ra) + A(sb) = r(Aa) + s(Ab) = 0 + 0 = 0.  This shows that ra + sb is a solution to the same homogeneous solution.