Editing Quotient space (linear algebra) (section)

== Properties ==

There is a natural [[epimorphism]] from ''V'' to the quotient space ''V''/''U'' given by sending ''x'' to its equivalence class [''x'']. The [[kernel (linear algebra)|kernel]] (or nullspace) of this epimorphism is the subspace ''U''. This relationship is neatly summarized by the [[short exact sequence]]
:<math>0\to U\to V\to V/U\to 0.\,</math>

If ''U'' is a subspace of ''V'', the [[dimension (vector space)|dimension]] of ''V''/''U'' is called the '''[[codimension]]''' of ''U'' in ''V''. Since a [[basis (linear algebra)|basis]] of ''V'' may be constructed from a basis ''A'' of ''U'' and a basis ''B'' of ''V''/''U'' by adding a [[representative (mathematics)|representative]] of each element of ''B'' to ''A'', the dimension of ''V'' is the sum of the dimensions of ''U'' and ''V''/''U''. If ''V'' is [[dimension (vector space)|finite-dimensional]], it follows that the codimension of ''U'' in ''V'' is the difference between the dimensions of ''V'' and ''U'':<ref>{{Harvard citation text|Axler|2015}} p. 97, § 3.89</ref><ref>{{Harvard citation text|Halmos|1974}} p. 34, § 22, Theorem 2</ref>
:<math>\mathrm{codim}(U) = \dim(V/U) = \dim(V) - \dim(U).</math>

Let ''T'' : ''V'' → ''W'' be a [[linear operator]]. The kernel of ''T'', denoted ker(''T''), is the set of all ''x'' in ''V'' such that ''Tx'' = 0. The kernel is a subspace of ''V''. The [[first isomorphism theorem]] for vector spaces says that the quotient space ''V''/ker(''T'') is isomorphic to the [[image (mathematics)|image]] of ''V'' in ''W''. An immediate [[corollary]], for finite-dimensional spaces, is the [[rank–nullity theorem]]: the dimension of ''V'' is equal to the dimension of the kernel (the [[kernel (linear algebra)|nullity]] of ''T'') plus the dimension of the image (the [[rank (linear algebra)|rank]] of ''T'').

The [[cokernel]] of a linear operator ''T'' : ''V'' → ''W'' is defined to be the quotient space ''W''/im(''T'').