162 CHAPTER 8. RANK OF A MATRIX

and so0 =∑

j ̸=ic jv j +(−1)vi,

contradicting the condition about the sum. ■Sometimes we refer to this last condition about sums as follows: The set of vectors,

{v1, · · · ,vk} is linearly independent if and only if there is no nontrivial linear combinationwhich equals zero. (A nontrivial linear combination is one in which not all the scalars equalzero.)

We give the following equivalent definition of linear independence which follows fromthe above corollaries.

Definition 8.5.4 A set of vectors, {v1, · · · ,vk} is linearly independent if and only if noneof the vectors is a linear combination of the others or equivalently if there is no nontriviallinear combination of the vectors which equals 0. It is said to be linearly dependent if atleast one of the vectors is a linear combination of the others or equivalently there exists anontrivial linear combination which equals zero.

Note the meaning of the words. To say a set of vectors is linearly dependent means atleast one is a linear combination of the others. In other words, it is in a sense “dependent”on these other vectors. At this time, the vectors are in Fn but the above definition makessense without knowing any description of the vectors. This will be considered later in thebook.

The following corollary follows right away from the row reduced echelon form. Itconcerns a matrix which looks like this: (More columns than rows.)

Corollary 8.5.5 Let {v1, · · · ,vk} be a set of vectors in Fn. Then if k > n, it must be the casethat {v1, · · · ,vk} is not linearly independent. In other words, if k > n, then {v1, · · · ,vk} isdependent.

Proof: If k > n, then the columns of(

v1 v2 · · · vk

)cannot each be a pivot

column because there are at most n pivot columns due to the fact the matrix has only nrows. In reading from left to right, pick the first column which is not a pivot column. Thenfrom the description of row reduced echelon form, this column is a linear combination ofthe preceding columns and so the given vectors are dependent by Corollary 8.5.2. ■

Example 8.5.6 Are the vectors



1230



2101



0112



322−1

 linearly indepen-

dent? If they are linearly dependent, exhibit one of the vectors as a linear combination ofthe others.