8.5. LINEAR INDEPENDENCE AND BASES 163

Form the matrix mentioned above.1 2 0 32 1 1 23 0 1 20 1 2 −1

Then the row reduced echelon form of this matrix is

1 0 0 10 1 0 10 0 1 −10 0 0 0

 .

Thus not all the columns are pivot columns and so the vectors are not linear independent.Note the fourth column is of the form

1

1000

+1

0100

+(−1)

0010

From Lemma 8.2.5, the same linear relationship exists between the columns of the originalmatrix. Thus

1

1230

+1

2101

+(−1)

0112

=

322−1

 .

Note the usefulness of the row reduced echelon form in discovering hidden linear rela-tionships in collections of vectors.

Example 8.5.7 Determine whether the vectors



1230



2101



0112



3220

 are

linearly independent. If they are linearly dependent, exhibit one of the vectors as a linearcombination of the others.

The matrix used to find this is 1 2 0 32 1 1 23 0 1 20 1 2 0

