kernel of a linear mapping
Let be a linear mapping between vector spaces.
The set of all vectors in that maps to is called the kernel (or nullspace) of , and is denoted . So
The kernel is a vector subspace of , and its dimension (http://planetmath.org/Dimension2) is called the nullity of .
The function is injective if and only if (see the attached proof (http://planetmath.org/OperatornamekerL0IfAndOnlyIfLIsInjective)). In particular, if the dimensions of and are equal and finite, then is invertible if and only if .
If is a vector subspace of , then we have
where is the restriction (http://planetmath.org/RestrictionOfAFunction) of to .
When the linear mappings are given by means of matrices, the kernel of the matrix is
Title | kernel of a linear mapping |
Canonical name | KernelOfALinearMapping |
Date of creation | 2013-03-22 11:58:22 |
Last modified on | 2013-03-22 11:58:22 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 20 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 15A04 |
Synonym | nullspace |
Synonym | null-space |
Synonym | kernel |
Related topic | LinearTransformation |
Related topic | ImageOfALinearTransformation |
Related topic | Nullity |
Related topic | RankNullityTheorem |