Editing Linear algebra (section)

===Linear maps===
{{main|Linear map}}
'''Linear maps''' are [[map (mathematics)|mappings]] between vector spaces that preserve the vector-space structure. Given two vector spaces {{math|''V''}} and {{math|''W''}} over a field {{mvar|F}}, a linear map (also called, in some contexts, linear transformation or linear mapping) is a [[map (mathematics)|map]]
: <math> T:V\to W </math>
that is compatible with addition and scalar multiplication, that is

: <math> T(\mathbf u + \mathbf v)=T(\mathbf u)+T(\mathbf v), \quad T(a \mathbf v)=aT(\mathbf v) </math>

for any vectors {{math|'''u''','''v'''}} in {{math|''V''}} and scalar {{math|''a''}} in {{mvar|F}}.

An equivalent condition is that for any vectors {{math|'''u''', '''v'''}} in {{math|''V''}} and scalars {{math|''a'', ''b''}} in {{mvar|F}}, one has

: <math>T(a \mathbf u + b \mathbf v) = aT(\mathbf u) + bT(\mathbf v) </math>.

When {{math|1=''V'' = ''W''}} are the same vector space, a linear map {{math|''T'' : ''V'' → ''V''}} is also known as a ''linear operator'' on {{mvar|V}}.

A [[bijective]] linear map between two vector spaces (that is, every vector from the second space is associated with exactly one in the first) is an [[isomorphism]]. Because an isomorphism preserves linear structure, two isomorphic vector spaces are "essentially the same" from the linear algebra point of view, in the sense that they cannot be distinguished by using vector space properties. An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its [[Range of a function|range]] (or image) and the set of elements that are mapped to the zero vector, called the [[Kernel (linear operator)|kernel]] of the map. All these questions can be solved by using [[Gaussian elimination]] or some variant of this [[algorithm]].