x^2-1=0

EXAMPLE 3 Determine if the following sets of vectors are linearly independent.

a.

b.

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SOLUTION

a. Notice that is a multiple of , namely . Hence , which shows that is linearly dependent.

b. The vectors and are certainly not multiples of one another. Could they be linearly dependent? Suppose and satisfy

If , then we can solve for in terms of , namely . This result is impossible because is not a multiple of . So must be zero. Similarly, must also be zero. Thus is a linearly independent set.

The arguments in Example 3 show that you can always decide by inspection when a set of two vectors is linearly dependent. Row operations are unnecessary. Simply check whether at least one of the vectors is a scalar times the other. (The test applies only to sets of two vectors.)

A set of two vectors is linearly dependent if at least one of the vectors is a multiple of the other. The set is linearly independent if and only if neither of the vectors is a multiple of the other.