rank-nullity theorem (original) (raw)

Note that if U is a subspace of V, then this (applied to the canonical mapping V→V/U) says that

that is,

where codim denotes codimension.

An alternative way of stating the rank-nullity theorem is by saying that if

is a short exact sequence of vector spaces, then

In fact, if

is an exact sequence of vector spaces, then

that is, the sum of the dimensions of even-numbered terms is the same as the sum of the dimensions of the odd-numbered terms.