Question

4. Suppose that we have a linear system given in matrix form as Ax = b, where A is an m×n matrix, b is an m×1 column vector, and x is an n×1 column vector. Suppose also that the n × 1 vector u is a solution to this linear system. Answer parts a. and b. below.

a. Suppose that the n × 1 vector h is a solution to the homogeneous linear system Ax=0. Showthenthatthevectory=u+hisasolutiontoAx=b.

b. Now, suppose that z is a solution to Ax = b: show that z can be written as z = u + h1, where h1 is a solution to the homogeneous system Ax = 0. (Note that h1 might not be the vector h from part a..)

Remark: Please take care to note that part a. and b. are asking slightly different questions: for example, in part a. you are being given a solution to the homogeneous system (h). In part b. you are deducing that such a solution (h1) must exist.

Answer 1