# kernel of a linear mapping

Let $T\colon V\to W$ be a linear mapping between vector spaces.

The set of all vectors in $V$ that $T$ maps to $0$ is called the kernel (or nullspace) of $T$, and is denoted $\ker T$. So

 $\ker T=\{\,x\in V\mid T(x)=0\,\}.$

The kernel is a vector subspace of $V$, and its dimension (http://planetmath.org/Dimension2) is called the nullity of $T$.

The function $T$ is injective if and only if $\ker T=\{0\}$ (see the attached proof (http://planetmath.org/OperatornamekerL0IfAndOnlyIfLIsInjective)). In particular, if the dimensions of $V$ and $W$ are equal and finite, then $T$ is invertible if and only if $\ker T=\{0\}$.

If $U$ is a vector subspace of $V$, then we have

 $\ker T|_{U}=U\cap\ker T,$

where $T|_{U}$ is the restriction (http://planetmath.org/RestrictionOfAFunction) of $T$ to $U$.

When the linear mappings are given by means of matrices, the kernel of the matrix $A$ is

 $\ker A=\{\,x\in V\mid Ax=0\,\}.$
 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