Editing Vector space (section)

===Eigenvalues and eigenvectors===
{{Main|Eigenvalues and eigenvectors}}
[[Endomorphism]]s, linear maps {{math|''f'' : ''V'' → ''V''}}, are particularly important since in this case vectors {{math|'''v'''}} can be compared with their image under {{math|''f''}}, {{math|''f''('''v''')}}. Any nonzero vector {{math|'''v'''}} satisfying {{math|1=''λ'''''v''' = ''f''('''v''')}}, where {{math|''λ''}} is a scalar, is called an ''eigenvector'' of {{math|''f''}} with ''eigenvalue'' {{math|''λ''}}.{{sfn|Roman|2005||loc=ch. 8, p. 135–156}} Equivalently, {{math|'''v'''}} is an element of the [[Kernel (linear algebra)|kernel]] of the difference {{math|''f'' − ''λ'' · Id}} (where Id is the [[identity function|identity map]] {{math|''V'' → ''V'')}}. If {{math|''V''}} is finite-dimensional, this can be rephrased using determinants: {{math|''f''}}  having eigenvalue {{math|''λ''}} is equivalent to
<math display=block>\det(f - \lambda \cdot \operatorname{Id}) = 0.</math>
By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in {{math|''λ''}}, called the [[characteristic polynomial]] of {{math|''f''}}.{{sfn||Lang|1987|loc=ch. IX.4}} If the field {{math|''F''}} is large enough to contain a zero of this polynomial (which automatically happens for {{math|''F''}} [[algebraically closed field|algebraically closed]], such as {{math|1=''F'' = '''C'''}}) any linear map has at least one eigenvector. The vector space {{math|''V''}} may or may not possess an [[eigenbasis]], a basis consisting of eigenvectors. This phenomenon is governed by the [[Jordan canonical form]] of the map.{{sfn|Roman|2005|loc=ch. 8, p. 140}} The set of all eigenvectors corresponding to a particular eigenvalue of {{math|''f''}} forms a vector space known as the ''eigenspace'' corresponding to the eigenvalue (and {{math|''f''}}) in question.