| Theorem |
|
1) Any two non-zero vectors are linear dependent if and only if they are collinear. 2) Any three non-zero vectors are linear dependent if and only if they are coplanar. 3) Any four vectors are linear dependent. |
Proof:
1) The equation
has a non-zero solution with respect to 
if and only if 
is the opposite vector of 
. It is possible only for collinear
vectors, and so they are linear dependent.
2) The equation
is equivalent to the following homogeneous system of the linear equations:
Assume that vectors 
are coplanar. If we take a rectangular
coordinate system such that the vectors lie in the x,y-plane
then
.
The remaining homogeneous system of two linear equations with three unknowns
has a non-zero solution (as if was
shown earlier). Therefore, a set of three coplanar vectors is linear dependent.
Now suppose that vectors 
are non-coplanar.
Note that any two vectors 
are coplanar, and any their linear
combination is a vector lying in the same plane. If 
does not lie in the same plane, then
it cannot be expressed as a linear combination of 
. Hence, a set of three non-coplanar
vectors is linear independent.
3) In case of four vectors we have the equation
,
which is equivalent to the homogeneous system of three linear equations
with four unknowns 
. Such system has an infinitely many
solutions, and so any set of four vectors is linear dependent.
