Editing Linear algebra (section)

===Subspaces, span, and basis===
{{main|Linear subspace|Linear span|Basis (linear algebra)}}
The study of those subsets of vector spaces that are in themselves vector spaces under the induced operations is fundamental, similarly as for many mathematical structures. These subsets are called [[linear subspace]]s. More precisely, a linear subspace of a vector space {{mvar|V}} over a field {{mvar|F}} is a [[subset]] {{mvar|W}} of {{mvar|V}} such that {{math|'''u''' + '''v'''}} and {{math|''a'''''u'''}} are in {{mvar|W}}, for every {{Math|'''u'''}}, {{Math|'''v'''}} in {{mvar|W}}, and every {{mvar|a}} in {{mvar|F}}. (These conditions suffice for implying that {{mvar|W}} is a vector space.)

For example, given a linear map {{math|''T'' : ''V'' → ''W''}}, the [[image (function)|image]] {{math|''T''(''V'')}} of {{mvar|V}}, and the [[inverse image]] {{math|''T''<sup>−1</sup>('''0''')}} of {{math|'''0'''}} (called [[kernel (linear algebra)|kernel]] or null space), are linear subspaces of {{mvar|W}} and {{mvar|V}}, respectively.

Another important way of forming a subspace is to consider [[linear combination]]s of a set {{mvar|S}} of vectors: the set of all sums 
: <math> a_1 \mathbf v_1 + a_2 \mathbf v_2 + \cdots + a_k \mathbf v_k,</math>
where {{math|'''v'''<sub>1</sub>, '''v'''<sub>2</sub>, ..., '''v'''<sub>''k''</sub>}} are in {{mvar|S}}, and {{math|''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''k''</sub>}} are in {{mvar|F}} form a linear subspace called the [[Linear span|span]] of {{mvar|S}}. The span of {{mvar|S}} is also the intersection of all linear subspaces containing {{mvar|S}}. In other words, it is the smallest (for the inclusion relation) linear subspace containing {{mvar|S}}.

A set of vectors is [[linearly independent]] if none is in the span of the others. Equivalently, a set {{mvar|S}} of vectors is linearly independent if the only way to express the zero vector as a linear combination of elements of {{mvar|S}} is to take zero for every coefficient {{mvar|a<sub>i</sub>}}.

A set of vectors that spans a vector space is called a [[spanning set]] or [[generating set]]. If a spanning set {{mvar|S}} is ''linearly dependent'' (that is not linearly independent), then some element {{Math|'''w'''}} of {{mvar|S}} is in the span of the other elements of {{mvar|S}}, and the span would remain the same if one were to remove {{Math|'''w'''}} from {{mvar|S}}. One may continue to remove elements of {{mvar|S}} until getting a ''linearly independent spanning set''. Such a linearly independent set that spans a vector space {{mvar|V}} is called a [[Basis (linear algebra)|basis]] of {{math|''V''}}. The importance of bases lies in the fact that they are simultaneously minimal-generating sets and maximal independent sets. More precisely, if {{mvar|S}} is a linearly independent set, and {{mvar|T}} is a spanning set such that {{math|''S'' ⊆ ''T''}}, then there is a basis {{mvar|B}} such that {{math|''S'' ⊆ ''B'' ⊆ ''T''}}.

Any two bases of a vector space {{math|''V''}} have the same [[cardinality]], which is called the [[Dimension (vector space)|dimension]] of {{math|''V''}}; this is the [[dimension theorem for vector spaces]]. Moreover, two vector spaces over the same field {{mvar|F}} are [[isomorphic]] if and only if they have the same dimension.<ref>{{Harvp|Axler|2015}} p. 82, §3.59</ref>

If any basis of {{math|''V''}} (and therefore every basis) has a finite number of elements, {{math|''V''}} is a ''finite-dimensional vector space''. If {{math|''U''}} is a subspace of {{math|''V''}}, then {{math|dim ''U'' ≤ dim ''V''}}. In the case where {{math|''V''}} is finite-dimensional, the equality of the dimensions implies {{math|1=''U'' = ''V''}}.

If {{math|''U''<sub>1</sub>}} and {{math|''U''<sub>2</sub>}} are subspaces of {{math|''V''}}, then

:<math>\dim(U_1 + U_2) = \dim U_1 + \dim U_2 - \dim(U_1 \cap U_2),</math>
where {{math|''U''<sub>1</sub> + ''U''<sub>2</sub>}} denotes the span of {{math|''U''<sub>1</sub> ∪ ''U''<sub>2</sub>}}.<ref>{{Harvp|Axler|2015}} p. 23, §1.45</ref>