Page 66 - A Course in Linear Algebra with Applications

50 Chapter Two: Systems of Linear Equations

are just the transposes of row echelon form and reduced row
echelon form respectively.
Example 2.3.3
/ 3 6 2 \
Put the matrix A = I J in reduced column echelon
form.
Apply the column operations \C\, C 2 — 6Ci, C3 — 2Ci,

C 2 <-> C 3 , ^ C 2 , and C x - \C 2 :

1 6 2 \ / 1 0 0
A
1/3 2 7J ^ \l/3 0 19/3

1 0 0 \ / 1 0 0
1/3 19/3 0y ~* ^ 1/3 1 0

1 0 0
0 1 0

We leave the reader to write down the elementary matrices
that produce these column operations.

Now suppose we are allowed to apply both row and column
operations to a matrix. Then we can obtain first row echelon
form; subsequently column operations may be applied to give
a matrix of the very simple type

0
'I r
0 0

where r is the number of pivots. This is called the normal
form of the matrix; we shall see in 5.2 that every matrix has
a unique normal form. These conclusions are summed up in
the following result.