Editing Linear subspace (section)

== Definition==
If ''V'' is a vector space over a [[Field (mathematics)|field]] ''K'', a subset ''W'' of ''V'' is a '''linear subspace''' of ''V'' if it is a [[vector space]] over ''K'' for the operations of ''V''. Equivalently, a linear subspace of ''V'' is a [[Empty set|nonempty]] subset ''W'' such that, whenever {{math|''w''<sub>1</sub>, ''w''<sub>2</sub>}} are elements of ''W'' and {{math|''α'', ''β''}} are elements of ''K'', it follows that {{math|''αw''<sub>1</sub> + ''βw''<sub>2</sub>}} is in ''W''.<ref>{{harvtxt|Anton|2005|p=155}}</ref><ref>{{harvtxt|Beauregard|Fraleigh|1973|p=176}}</ref><ref>{{harvtxt|Herstein|1964|p=132}}</ref><ref>{{harvtxt|Kreyszig|1972|p=200}}</ref><ref>{{harvtxt|Nering|1970|p=20}}</ref>

The [[singleton set]] consisting of the [[zero vector]] alone and the entire vector space itself are linear subspaces that are called the '''trivial subspaces''' of the vector space.<ref>{{harvtxt|Hefferon|2020}} p. 100, ch. 2, Definition 2.13</ref>