Linearly Independence and Dependence.

Linearly independent vectors:

A set of vectors {v1,v2,..vN}are said to be linearly independent if  a1v1+a2v2+...aNvN  = 0  implies  ai=0 for all i=1,2,...N. Otherwise they are said to be linearly dependent vectors.

Example:

1) Consider the vector space V=R over R.

 Here {1} , {2} , {100001} are all some linearly independent sets.

 One can check that {1,5} , {1,2} , {0,1} are not linearly independent sets.

2) Consider the vector space V=R2 over R

 Here {(1,0),(0,1)} , {(1,0),(1,1)} are some linearly independent sets.

 One can check that {(1,0),(5,0)} , {(1,0),(0,0)} are not linearly independent.

Observe that when the set contains null vector,then the set is always a linearly dependent set.If one vector belongs to linear span of some other vector of the same set,then the set is linearly dependent set.The maximum number of linearly independent vectors in a finite dimensional spaces is equal to dimension of the vector space.

A set X is linearly independent if every finite number of vectors of set X are linearly independent.Otherwise linearly dependent.