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