Editing Linear independence (section)

=== Linear dependence and independence of two vectors ===

This example considers the special case where there are exactly two vector <math>\mathbf{u}</math> and <math>\mathbf{v}</math> from some real or complex vector space. 
The vectors <math>\mathbf{u}</math> and <math>\mathbf{v}</math> are linearly dependent [[if and only if]] at least one of the following is true: 
# <math>\mathbf{u}</math> is a scalar multiple of <math>\mathbf{v}</math> (explicitly, this means that there exists a scalar <math>c</math> such that <math>\mathbf{u} = c \mathbf{v}</math>) or 
# <math>\mathbf{v}</math> is a scalar multiple of <math>\mathbf{u}</math> (explicitly, this means that there exists a scalar <math>c</math> such that <math>\mathbf{v} = c \mathbf{u}</math>). 
If <math>\mathbf{u} = \mathbf{0}</math> then by setting <math>c := 0</math> we have <math>c \mathbf{v} = 0 \mathbf{v} = \mathbf{0} = \mathbf{u}</math> (this equality holds no matter what the value of <math>\mathbf{v}</math> is), which shows that (1) is true in this particular case. Similarly, if <math>\mathbf{v} = \mathbf{0}</math> then (2) is true because <math>\mathbf{v} = 0 \mathbf{u}.</math> 
If <math>\mathbf{u} = \mathbf{v}</math> (for instance, if they are both equal to the zero vector <math>\mathbf{0}</math>) then ''both'' (1) and (2) are true (by using <math>c := 1</math> for both).

If <math>\mathbf{u} = c \mathbf{v}</math> then <math>\mathbf{u} \neq \mathbf{0}</math> is only possible if <math>c \neq 0</math> ''and'' <math>\mathbf{v} \neq \mathbf{0}</math>; in this case, it is possible to multiply both sides by <math display="inline">\frac{1}{c}</math> to conclude <math display="inline">\mathbf{v} = \frac{1}{c} \mathbf{u}.</math> 
This shows that if <math>\mathbf{u} \neq \mathbf{0}</math> and <math>\mathbf{v} \neq \mathbf{0}</math> then (1) is true if and only if (2) is true; that is, in this particular case either both (1) and (2) are true (and the vectors are linearly dependent) or else both (1) and (2) are false (and the vectors are linearly ''in''dependent). 
If <math>\mathbf{u} = c \mathbf{v}</math> but instead <math>\mathbf{u} = \mathbf{0}</math> then at least one of <math>c</math> and <math>\mathbf{v}</math> must be zero. 
Moreover, if exactly one of <math>\mathbf{u}</math> and <math>\mathbf{v}</math> is <math>\mathbf{0}</math> (while the other is non-zero) then exactly one of (1) and (2) is true (with the other being false).

The vectors <math>\mathbf{u}</math> and <math>\mathbf{v}</math> are linearly ''in''dependent if and only if <math>\mathbf{u}</math> is not a scalar multiple of <math>\mathbf{v}</math> ''and'' <math>\mathbf{v}</math> is not a scalar multiple of <math>\mathbf{u}</math>.