Editing Linear subspace (section)

=== Example II ===
Let the field be '''R''' again, but now let the vector space ''V'' be the [[Cartesian plane]] '''R'''<sup>2</sup>.
Take ''W'' to be the set of points (''x'', ''y'') of '''R'''<sup>2</sup> such that ''x'' = ''y''.
Then ''W'' is a subspace of '''R'''<sup>2</sup>.

''Proof:''
#Let {{nowrap|1='''p''' = (''p''<sub>1</sub>, ''p''<sub>2</sub>)}} and {{nowrap|1='''q''' = (''q''<sub>1</sub>, ''q''<sub>2</sub>)}} be elements of ''W'', that is, points in the plane such that ''p''<sub>1</sub> = ''p''<sub>2</sub> and ''q''<sub>1</sub> = ''q''<sub>2</sub>. Then {{nowrap|1='''p''' + '''q''' = (''p''<sub>1</sub>+''q''<sub>1</sub>, ''p''<sub>2</sub>+''q''<sub>2</sub>)}}; since ''p''<sub>1</sub> = ''p''<sub>2</sub> and ''q''<sub>1</sub> = ''q''<sub>2</sub>, then ''p''<sub>1</sub> + ''q''<sub>1</sub> = ''p''<sub>2</sub> + ''q''<sub>2</sub>, so '''p''' + '''q''' is an element of ''W''.
#Let '''p''' = (''p''<sub>1</sub>, ''p''<sub>2</sub>) be an element of ''W'', that is, a point in the plane such that ''p''<sub>1</sub> = ''p''<sub>2</sub>, and let ''c'' be a scalar in '''R'''. Then {{nowrap|1=''c'''''p''' = (''cp''<sub>1</sub>, ''cp''<sub>2</sub>)}}; since ''p''<sub>1</sub> = ''p''<sub>2</sub>, then ''cp''<sub>1</sub> = ''cp''<sub>2</sub>, so ''c'''''p''' is an element of ''W''.

In general, any subset of the real coordinate space '''R'''<sup>''n''</sup> that is defined by a [[homogeneous system of linear equations]] will yield a subspace.
(The equation in example I was ''z''&nbsp;=&nbsp;0, and the equation in example II was ''x''&nbsp;=&nbsp;''y''.)