# Linear dependence and independence.

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

Definition 5.2.1   A family

of vectors in V is linearly independent if, for any scalars , we have:

 (5.1)

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

Example 5.2.2   We prove that the vectors and are linearly independent in .

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

Therefore:

Proposition 5.2.3   Let F be a family of vectors in V. If F is linearly independent, then any non empty subset of F is linearly independent too.

Definition 5.2.4   If a family F of vectors in V is not linearly independent, it is linearly dependent.

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

Example 5.2.5

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 .

Noah Dana-Picard
2001-02-26