Editing Linear subspace (section)

===Intersection===
[[File:Intersecting Planes 2.svg|thumb|right|In '''R'''<sup>3</sup>, the intersection of two distinct two-dimensional subspaces is one-dimensional]]
Given subspaces ''U'' and ''W'' of a vector space ''V'', then their [[intersection (set theory)|intersection]] ''U''&nbsp;∩&nbsp;''W''&nbsp;:= {'''v'''&nbsp;∈&nbsp;''V''&nbsp;: '''v'''&nbsp;is an element of both ''U'' and&nbsp;''W''} is also a subspace of ''V''.<ref>{{harvtxt|Nering|1970|p=21}}</ref>

''Proof:''
# Let '''v''' and '''w''' be elements of ''U''&nbsp;∩&nbsp;''W''. Then '''v''' and '''w''' belong to both ''U'' and ''W''. Because ''U'' is a subspace, then '''v'''&nbsp;+&nbsp;'''w''' belongs to ''U''. Similarly, since ''W'' is a subspace, then '''v'''&nbsp;+&nbsp;'''w''' belongs to ''W''. Thus, '''v'''&nbsp;+&nbsp;'''w''' belongs to ''U''&nbsp;∩&nbsp;''W''.
# Let '''v''' belong to ''U''&nbsp;∩&nbsp;''W'', and let ''c'' be a scalar. Then '''v''' belongs to both ''U'' and ''W''. Since ''U'' and ''W'' are subspaces, ''c'''''v''' belongs to both ''U'' and&nbsp;''W''.
# Since ''U'' and ''W'' are vector spaces, then '''0''' belongs to both sets. Thus, '''0''' belongs to ''U''&nbsp;∩&nbsp;''W''.

For every vector space ''V'', the [[zero vector space|set {'''0'''}]] and ''V'' itself are subspaces of ''V''.<ref>{{harvtxt|Hefferon|2020}} p. 100, ch. 2, Definition 2.13</ref><ref>{{harvtxt|Nering|1970|p=20}}</ref>