Editing Linear subspace (section)

==Operations and relations on subspaces==

=== Inclusion ===
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The [[inclusion relation|set-theoretical inclusion]] binary relation specifies a [[partial order]] on the set of all subspaces (of any dimension).

A subspace cannot lie in any subspace of lesser dimension. If dim&nbsp;''U''&nbsp;=&nbsp;''k'', a finite number, and ''U''&nbsp;⊂&nbsp;''W'', then dim&nbsp;''W''&nbsp;=&nbsp;''k'' if and only if ''U''&nbsp;=&nbsp;''W''.

===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>

===Sum===
If ''U'' and ''W'' are subspaces, their '''sum''' is the subspace<ref>{{harvtxt|Nering|1970|p=21}}</ref><ref name=":1">Vector space related operators.</ref>
<math display="block">U + W = \left\{ \mathbf{u} + \mathbf{w} \colon \mathbf{u}\in U, \mathbf{w}\in W \right\}.</math>

For example, the sum of two lines is the plane that contains them both. The dimension of the sum satisfies the inequality
<math display="block">\max(\dim U,\dim W) \leq \dim(U + W) \leq \dim(U) + \dim(W).</math>

Here, the minimum only occurs if one subspace is contained in the other, while the maximum is the most general case. The dimension of the intersection and the sum are related by the following equation:<ref>{{harvtxt|Nering|1970|p=22}}</ref>
<math display="block">\dim(U+W) = \dim(U) + \dim(W) - \dim(U \cap W).</math>

A set of subspaces is '''independent''' when the only intersection between any pair of subspaces is the trivial subspace. The '''[[Direct sum of modules|direct sum]]''' is the sum of independent subspaces, written as <math>U \oplus W</math>. An equivalent restatement is that a direct sum is a subspace sum under the condition that every subspace contributes to the span of the sum.<ref>{{harvtxt|Hefferon|2020}} p. 148, ch. 2, §4.10</ref><ref>{{harvtxt|Axler|2015}} p. 21 § 1.40</ref><ref>{{harvtxt|Katznelson|Katznelson|2008}} pp. 10–11, § 1.2.5</ref><ref>{{harvtxt|Halmos|1974}} pp. 28–29, § 18</ref>

The dimension of a direct sum <math>U \oplus W</math> is the same as the sum of subspaces, but may be shortened because the dimension of the trivial subspace is zero.<ref>{{harvtxt|Halmos|1974}} pp. 30–31, § 19</ref>

<math display="block">\dim (U \oplus W) = \dim (U) + \dim (W)</math>

=== Lattice of subspaces ===
The operations [[#Intersection|intersection]] and [[#Sum|sum]] make the set of all subspaces a bounded [[modular lattice]], where the [[zero vector space|{0} subspace]], the [[least element]], is an [[identity element]] of the sum operation, and the identical subspace ''V'', the greatest element, is an identity element of the intersection operation.

=== Orthogonal complements ===

If <math>V</math> is an [[inner product space]] and <math>N</math> is a subset of <math>V</math>, then the [[orthogonal complement]] of <math>N</math>, denoted <math>N^{\perp}</math>, is again a subspace.<ref>{{harvtxt|Axler|2015}} p. 193, § 6.46</ref> If <math>V</math> is finite-dimensional and <math>N</math> is a subspace, then the dimensions of <math>N</math> and <math>N^{\perp}</math> satisfy the complementary relationship <math>\dim (N) + \dim (N^{\perp}) = \dim (V) </math>.<ref>{{harvtxt|Axler|2015}} p. 195, § 6.50</ref>  Moreover, no vector is orthogonal to itself, so <math> N \cap N^\perp = \{ 0 \}</math> and <math>V</math> is the [[direct sum]] of <math>N</math> and <math>N^{\perp}</math>.<ref>{{harvtxt|Axler|2015}} p. 194, § 6.47</ref>  Applying orthogonal complements twice returns the original subspace: <math>(N^{\perp})^{\perp} = N</math> for every subspace <math>N</math>.<ref>{{harvtxt|Axler|2015}} p. 195, § 6.51</ref>

This operation, understood as [[negation]] (<math>\neg</math>), makes the lattice of subspaces a (possibly [[infinite set|infinite]]) orthocomplemented lattice (although not a distributive lattice).{{citation needed|date=January 2019}}

In spaces with other [[bilinear form]]s, some but not all of these results still hold. In [[pseudo-Euclidean space]]s and [[symplectic vector space]]s, for example, orthogonal complements exist. However, these spaces may have [[null vector]]s that are orthogonal to themselves, and consequently there exist subspaces <math>N</math> such that <math>N \cap N^{\perp} \ne \{ 0 \}</math>. As a result, this operation does not turn the lattice of subspaces into a Boolean algebra (nor a [[Heyting algebra]]).{{citation needed|date=January 2019}}