Editing Linear independence (section)

== Geometric examples ==
[[File:Vectores independientes.png|right]]

* <math>\vec u</math> and <math>\vec v</math> are independent and define the [[plane (geometry)|plane]] P.
* <math>\vec u</math>, <math>\vec v</math> and <math>\vec w</math> are dependent because all three are contained in the same plane.
* <math>\vec u</math> and <math>\vec j</math> are dependent because they are parallel to each other.
* <math>\vec u</math> , <math>\vec v</math> and <math>\vec k</math> are independent because <math>\vec u</math> and <math>\vec v</math> are independent of each other and <math>\vec k</math> is not a linear combination of them or, equivalently, because they do not belong to a common plane. The three vectors define a three-dimensional space.
* The vectors <math>\vec o</math> (null vector, whose components are equal to zero) and <math>\vec k</math> are dependent since <math>\vec o = 0 \vec k</math>.

=== Geographic location ===

A person describing the location of a certain place might say, "It is 3 miles north and 4 miles east of here."  This is sufficient information to describe the location, because the geographic coordinate system may be considered as a 2-dimensional vector space (ignoring altitude and the curvature of the Earth's surface).  The person might add, "The place is 5 miles northeast of here."  This last statement is ''true'', but it is not necessary to find the location.    
    
In this example the "3 miles north" vector and the "4 miles east" vector are linearly independent.  That is to say, the north vector cannot be described in terms of the east vector, and vice versa.  The third "5 miles northeast" vector is a [[linear combination]] of the other two vectors, and it makes the set of vectors ''linearly dependent'', that is, one of the three vectors is unnecessary to define a specific location on a plane.

Also note that if altitude is not ignored, it becomes necessary to add a third vector to the linearly independent set.  In general, {{mvar|n}} linearly independent vectors are required to describe all locations in {{mvar|n}}-dimensional space.