We consider a vector space *V* and a set *I* (this will be a set of indices, generally of the form
,
but not necessarily).

If *I* is a finite set, e.g.
,
we can write
Eq. 7 as follows:

Let be two scalars (i.e. here two real numbers). Then:

Therefore:

This means that there exists a family of scalars such that at least one of them is not zero and .

- 1.
- The family of one vector is linearly dependent, as .
- 2.
- Every family containing the zero vector is linearly dependent (take 1 as the coefficient of and zeros as the other coefficients).
- 3.
- The vectors and are linearly dependent in , as .