|
The linear system in (2.5)
is a particular example. We would like to begin considering linear systems
in general, so let's replace the specific coefficients of the example
with some notation that will allow us to study any linear system:
To get the system in (2.5), we let a11=1, a12=1, a13=2, a21=2, a22=0, a23=1, a31=1, a32=2, a33=1, b1=6, b2=4, and b3=7. In (2.6), we have called the unknowns x1, x2, and x3 because this notation is easier to extend to larger problems. In matrix form, (2.6) becomes 
Now suppose that we have a system of n equations in n unknowns.
We can write this system as
The dots indicate places where terms or equations following the same pattern exist but have not been written down. The variables are x1, x2,…, xn-1, xn, and we often write this as xi, for i = 1, 2, …,n.
The coefficient of variable xj in equation i is denoted aij, and the right-hand-side value in equation i is denoted bi. Note that if we let n=3 in (), then we get the systems in (2.6). We can write the system (2.7) in matrix form as
 If we let A denote the coefficient matrix, x the vector of unknowns, and b the vector of right-hand-side values, then the general linear system described above may be written compactly as Ax = b. The augmented matrix associated with the system is  which we can also write as [A | b]. |
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Updated: February 22, 2001
Copyright © 2001 Richard Tapia and Cynthia Lanius
