kernel of a linear mapping

Let T:VW be a linear mapping between vector spacesMathworldPlanetmath.

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


The kernel is a vector subspace of V, and its dimensionPlanetmathPlanetmath ( is called the nullityMathworldPlanetmath of T.

The function T is injective if and only if kerT={0} (see the attached proof ( In particular, if the dimensions of V and W are equal and finite, then T is invertiblePlanetmathPlanetmath if and only if kerT={0}.

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


where T|U is the restriction ( of T to U.

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