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:

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 .

2001-02-26