Solving linear systems

To solve Ax=b

1. Form the matrix M that represents forward elimination such that MA = U, where U is an upper triangular matrix
   if using method '2b' below, also form M^-1 and call it L, a lower triangular matrix.
   The lecture slides cover how to form M and M^-1

2. Substitute y for Ux and solve for y using one of the two methods below (note that in Section 8.1 of the 5th edition of the book, they use 'z' instead of 'y').
   • Option 1: Since Ax = M^-1MAx = LUx = b and y = Ux
     solve Ly=b for y using back substitution
     (Section 8.1, p. 321, Equation 20)
   • Option 2: Since MAx= Ux = Mb, and Ux = y
     y = Mb
     (Section 8.1, p. 322, 3rd equation from the top)

3. Now that y is known, solve Ux = y for x using back sustitution