DEFINITION.– We say that n vectors X1, …, Xn of L2 (dP) are linearly independent if a.s. (here 0 is the zero vector of L2 (dP)).
DEFINITION.– We say that the n vectors X1, …, X2 of L2 (dP) are linearly dependent if ∃ λ1, …,λn are not all zero and ∃ an event A of positive probability such that λ1X1 (ω) +… + λnXn (ω) = 0 ∀ω∈ A.
In particular: X1, …, Xn will be linearly dependent if ∃ λ1, …,λn are not all zero such that λ1 X1 +… + λn Xn = 0 a.s.
Examples: given the three measurable mappings:
The three mappings are evidently measurable and belong to L2(dω), so there are 3 vectors of L2(dω).
There 3 vectors are linearly dependent on A = [0,1[ of probability measurement :
Covariance matrix and linear ...