Editing Linear map (section)

==Change of basis==
{{Main|Basis (linear algebra)|Change of basis}}
Given a linear map which is an [[endomorphism]] whose matrix is ''A'', in the basis ''B'' of the space it transforms vector coordinates [u] as [v] = ''A''[u]. As vectors change with the inverse of ''B'' (vectors coordinates are [[Covariance and contravariance of vectors|contravariant]]) its inverse transformation is [v] = ''B''[v'].

Substituting this in the first expression
<math display="block">B\left[v'\right] = AB\left[u'\right]</math>
hence
<math display="block">\left[v'\right] = B^{-1}AB\left[u'\right] = A'\left[u'\right].</math>

Therefore, the matrix in the new basis is ''A′'' = ''B''<sup>−1</sup>''AB'', being ''B'' the matrix of the given basis.

Therefore, linear maps are said to be 1-co- 1-contra-[[covariance and contravariance of vectors|variant]] objects, or type (1, 1) [[tensor]]s.