Page 34 - Numerical Methods for Chemical Engineering

Existence and uniqueness of solutions 23

in the third row. We thus swap rows 2 and 3,
 
3 1 6 2
¯ ¯ (3, 1)
(A, b) =  0 2 3 −13 1  (1.117)

3 
0 1 −15 2
3 3
and perform a row operation to zero the (3, 2) element.

1 2 1
λ 32 = = (1.118)
3 3 2
 
3 1 6 2
(A, b) (3, 2) =  0 2 3 −1 3 1 3 


1
2
1 1 1 2 1
0 − −1 − (−1) 5 − 3
3 2 3 2 3 2 3
 
3 1 6 2
 0 2 −1 3 1  (1.119)
3 3 
= 
0 0 − 1 4
2
We now have an upper triangular system to solve by backward substitution,
3x 1 + x 2 + 6x 3 = 2
2 x 2 − x 3 = 3 1 (1.120)
3 3
1
− x 3 = 4
2
First, x 3 =−8 from the last equation. Then, from the second equation,
1 2

x 2 = 3 + x 3 =−7 (1.121)
3 3
Finally, from the first equation,
x 1 = (2 − 6x 3 − x 2 )/3 = 19 (1.122)
The solution to (1.70) is thus (x 1 , x 2 , x 3 ) = (19, −7, −8).
Existence and uniqueness of solutions

With Gaussian elimination and partial pivoting, we have a method for solving linear systems
that either finds a solution or fails under conditions in which no unique solution exists. In
this section, we consider at more depth the question of when a linear system possesses a real
solution (existence) and if so, whether there is exactly one (uniqueness). These questions are
vitally important, for linear algebra is the basis upon which we build algorithms for solving
nonlinear equations, ordinary and partial differential equations, and many other tasks.

Interpreting Ax = b as a linear transformation
As a first step, we consider the equation Ax = b from a somewhat more abstract viewpoint.
We note that A is an N × N real matrix and x and b are both N-dimensional real vectors.