rank-nullity theorem
Let and be vector spaces over the same field. If is a linear mapping, then
In other words, the dimension of is equal to the sum (http://planetmath.org/CardinalArithmetic) of the rank (http://planetmath.org/RankLinearMapping) and nullity of .
Note that if is a subspace of , then this (applied to the canonical mapping ) says that
that is,
where 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.
Title | rank-nullity theorem |
---|---|
Canonical name | RanknullityTheorem |
Date of creation | 2013-03-22 16:35:40 |
Last modified on | 2013-03-22 16:35:40 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 7 |
Author | yark (2760) |
Entry type | Theorem |
Classification | msc 15A03 |
Related topic | RankLinearMapping |
Related topic | Nullity |