Lecture 20
February 23. 2007
Definition
Two systems of linear equations are called equivalent if they have
the same solution set .
Theorem
Let Ax = b be a system of m linear equations in n unknowns , and
let C be an invertible m × m matrix. Then the system (CA)x = Cb
is equivalent to Ax = b.
Corollary
Let Ax = b be a system of m linear equations in n unknowns. If
(A'\b') is obtained from (A\b) by a finite number of elementary row
operations , then the system A'x = b' is equivalent to the original
system.
Definition
A matrix is said to be in reduced row echelon form if the
fol lowing three conditions are satisfied:
1 Any rows containing a non zero entry orecedes any row in
which all the entries are zero (if any).
2 The first nonzero entry in each row is the only nonzero entry
in its column.
3 The first nonzero entry in each row is 1 and it occurs in a
column to the right of the first nonzero entry in the preceding
row.
Theorem
Gaussian elimination transforms any matrix into its reduced row
echelon form.
