kernel of a linear mapping

Let $T:V\to W$ 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