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# 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 .    Next: Generating set of a Up: Vector spaces. Previous: Vector spaces and vector
Noah Dana-Picard
2001-02-26