# Linear Dependence and Linear Independence Vectors

__Linear Dependence__

Let be a vector space and let be a finite subset of . Then is said to be linearly dependent if there exists scalar , not all zero, such that

__Linear Independence__

Let be a vector space and let be a finite subset of . Then is said to be linearly independent if,

This holds only when .

An infinite subset of is said to be linearly independent if every finite subset is linearly independent, otherwise it is linearly dependent.

__Example 1__**:** Show that the system of three vectors , , of is linearly dependent.

__Solution__**:** For .

Therefore, the given system of vectors is linearly dependent.

__Example 2__**:** Consider the vector space and the subset of . Prove that is linearly independent.

__Solution__**:** For .

This shows that if any linear combination of the elements of is zero then the coefficient must be zero. is linearly independent.