7.4 SIMULTANEOUS LINEAR EQUATIONS

We now describe methods of solving systems of linear equations which are written as

(7.4 1a,b,c)

A more compact form of writing Equations (7.4 1a,b,c) is

(7.4 2a)

or

(7.4 2b)

where =(a _ij) is the coefficient matrix, and x and b are column matrices. Equation (7.4 2b) has a unique solution if the inverse of exists. The conditions for the existence of ^?1 can be stated in the following alternate forms.

1. The determinant of is not zero.

2. The columns of are linearly independent.

3. The rows of are linearly independent.

4. The homogenous system ( x=0) has only the trivial solution ( x=0).

5. The rank of is n.

If ^?1 exists, we say that the matrix is non-singular; otherwise is said to be a singular matrix. Thus the linear system (7.4 2b) has a unique solution iff is non-singular. Recall also that if { v ₁, v ₂, , v _n} is a set of vectors in R ⁿ, the following statements are equivalent.

1. Vectors { v ₁, v ₂, , v _n} are linearly independent.

2. Vectors { v ₁, v ₂, , v _n} span R ⁿ.

3. Vectors { v ₁, v ₂, , v _n) form a basis for R ⁿ.

We now discuss a standard method of...