Example – 4
Suppose that for three non-zero vectors  , , , any two of them are non-collinear. If the vectors ( + 2 )and
are collinear and the vectors ( + 3 )
and are collinear, prove that
+ 2 + 6 = 
Solution: We must have some l,m Î R such that
+ 2 =l ... (1)
+ 3= l... (2)
From (1), we have
= (1/l ) ( + 2) …..(3)
We use this in (2) :
Since and
are non-collinear, their linear combination can be zero if and only if the two scalars
are zero. This gives
(3/l) -m = 0
1+ (6/l) = 0
Þ l =- 6, m = -(1/2)
Using the value of l in (3), we have
+ 2 + 6 = 0
In the preceeding discussion, we talked about the basis of a plane. We can easily extend that discussion to observe that any three non-coplanar vectors can form a basis of three dimensional space:
In other words, any vector  in 3-D space can be expressed as a linear combination of three arbitrary non-coplanar vectors. From this, it also follows that for three non-coplanar vectors , , , if their linear combination is zero, i.e, if
l+ m+g = (where l,m,g ÎR)
thenl,m and g 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, ë,ì and ã must be zero.
We finally come to what we mean by linearly independent and linearly dependent vectors.
Linearly independent : A set of non-zero vectors 1, 2 , 3 , ….. , n is said to be linearly independent if
Vectors
implies l1 = l2 = …..= ln = 0
Thus, a linear combination of linearly independent vectors cannot be zero unless all the
scalars used to form the linear combination are zero.
Linearly dependent A set of non-zero vectors 1 , 2 , 3,…. n is said to be linearly dependent if there exist
vectors: scalars l1 , l2 ,….., ln not all zero such that,
l11 + l22 + lnn =
For example, based on our previous discussions, we see that
(i) Two non-zero, non-collinear vectors are linearly independent.
(ii) Two collinear vectors are linearly dependent
(iii) Three non-zero, non-coplanar vectors are linearly independent.
(iv) Three coplanar vectors are linearly dependent
(v) Any four vectors in 3-D space are linearly dependent.
You are urged to prove for yourself all these assertions.
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