In the preceeding discussion, we talked about the basis of a plane. We can easily extend that discussion to observe that any three noncoplanar vectors can form a basis of three dimensional space:
In other words, any vector in space can be expressed as a linear combination of three arbitrary noncoplanar vectors. From this, it also follows that for three noncoplanar vectors if their linear combination is zero, i.e, if
where 
then and must all be zero. To prove this, assume the contrary. Then, we have
which means that can be written as the linear combination of and . However, this would make and coplanar, contradicting our initial supposition. Thus, and must be zero.
We finally come to what we mean by linearly independent and linearly dependent vectors.
Linearly independent vectors:
A set of nonzero vectors is said to be linearly independent if
implies 
Thus, a linear combination of linearly independent vectors cannot be zero unless all the scalars used to form the linear combination are zero.
Linear dependent vectors:
A set of nonzero vectors is said to be linearly dependent if there exist scalars not all zero such that,
For example, based on our previous discussions, we see that
(i) Two nonzero, noncollinear vectors are linearly independent.  
(ii) Two collinear vectors are linearly dependent  
(iii) Three nonzero, noncoplanar vectors are linearly independent.  
(iv) Three coplanar vectors are linearly dependent  
(v) Any four vectors in space are linearly dependent. 
You are urged to prove for yourself all these assertions.
Example: 5  
Let and be noncoplanar vectors. Are the vectors and coplanar or noncoplanar?

Solution: 5  
Three vectors are coplanar if there exist scalars using which one vector can be expressed as the linear combination of the other two.
Let us try to find such scalars:
Since are noncoplanar, we must have
This system, as can be easily verified , does not have a solution for and .
Thus, we cannot find scalars for which one vector can be expressed as the linear combination of the other two, implying the three vectors must be noncoplanar.
As an additional exercise, show that for three noncoplanar vectors and , the vectors and are coplanar.

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