**1.5. Linear independence of vectors of** *L*^{2} (*dP*)

DEFINITION.– We say that *n* vectors *X*_{1}, …, *X*_{n} of *L*^{2} (*dP*) are linearly independent if a.s. (here 0 is the zero vector of *L*^{2} (*dP*)).

DEFINITION.– We say that the *n* vectors *X*_{1}, …, *X*_{2} of *L*^{2} (*dP*) are linearly dependent if ∃ λ_{1}, …,λ_{n} are not all zero and ∃ an event *A* of positive probability such that λ_{1}*X*_{1} (*ω*) +… + λ_{n}*X*_{n} (*ω*) = 0 ∀*ω*∈ *A*.

In particular: *X*_{1}, …, *X*_{n} will be linearly dependent if ∃ λ_{1}, …,λ_{n} are not all zero such that λ_{1} *X*_{1} +… + λ_{n} *X*_{n} = 0 a.s.

Examples: given the three measurable mappings:

defined by:

**Figure 1.6.** *Three random variables*

The three mappings are evidently measurable and belong to *L*^{2}(*dω*), so there are 3 vectors of *L*^{2}(*dω*).

There 3 vectors are linearly dependent on *A* = [0,1[ of probability measurement :

**Covariance matrix and linear ...**