Editing Linear independence (section)

===Linearly independent vector subspaces===

Two vector subspaces <math>M</math> and <math>N</math> of a vector space <math>X</math> are said to be {{em|linearly independent}} if <math>M \cap N = \{0\}.</math><ref name="BNFA">{{Bachman Narici Functional Analysis 2nd Edition}} pp. 3–7</ref> 
More generally, a collection <math>M_1, \ldots, M_d</math> of subspaces of <math>X</math> are said to be {{em|linearly independent}} if <math display=inline>M_i \cap \sum_{k \neq i} M_k = \{0\}</math> for every index <math>i,</math> where <math display=inline>\sum_{k \neq i} M_k = \Big\{m_1 + \cdots + m_{i-1} + m_{i+1} + \cdots + m_d : m_k \in M_k \text{ for all } k\Big\} = \operatorname{span} \bigcup_{k \in \{1,\ldots,i-1,i+1,\ldots,d\}} M_k.</math><ref name="BNFA" />
The vector space <math>X</math> is said to be a {{em|[[direct sum]]}} of <math>M_1, \ldots, M_d</math> if these subspaces are linearly independent and <math>M_1 + \cdots + M_d = X.</math>