Editing Coordinate vector (section)

==Infinite-dimensional vector spaces==

Suppose ''V'' is an infinite-dimensional vector space over a field ''F''. If the dimension is ''κ'', then there is some basis of ''κ'' elements for ''V''. After an order is chosen, the basis can be considered an ordered basis. The elements of ''V'' are finite linear combinations of elements in the basis, which give rise to unique coordinate representations exactly as described before. The only change is that the indexing set for the coordinates is not finite. Since a given vector ''v'' is a ''finite'' linear combination of basis elements, the only nonzero entries of the coordinate vector for ''v'' will be the nonzero coefficients of the linear combination representing ''v''. Thus the coordinate vector for ''v'' is zero except in finitely many entries.

The linear transformations between (possibly) infinite-dimensional vector spaces can be modeled, analogously to the finite-dimensional case, with [[Infinite matrix#Infinite matrices|infinite matrices]]. The special case of the transformations from ''V'' into ''V'' is described in the [[full linear ring]] article.