Hint:
)
)
)
Therefore, to be a linearly dependant set of vectors, then:
That condition isn't necessary for linear dependence; for example, if
the set is linearly dependent (because it contains the zero vector), but the above equation doesn't hold if
even though the set is dependent. The proper condition for dependence is that at least one of the following is true:
This is somewhat cumbersome since you generally would need to check each and every vector in the set, so we have the more symmetrical equivalent condition:
The vectors
are linearly dependent if and only if
for some scalars
which are not all zero. A similar definition applies for an arbitrary number of vectors.
This is more useful when we actually know the vectors, since by equating components we get three simultaneous equations which we can attempt to solve for
. It doesn't particularly help for this question, I just thought I'd point out that ahat's test won't always work unless you test every vector.
For this question, all you really need is that if a set of vectors contains two of the same vector, then it is linearly dependent. This is because, if we have the vectors
, we can simply write
, which satisfies the definition of linear dependence since not all of the coefficients are zero.
From there, notice that all three vectors are the same if
, so they will be linearly dependent for this value. It's not necessarily the only value, but I'm guessing that was a multiple-choice question.
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