Editing Linear subspace (section)

===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>