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kernel of a linear mapping (Definition)

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

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

The kernel is a vector subspace of $ V$, and its dimension is called the nullity of $ T$.

The function $ T$ is injective if and only if $ \ker T=\{0\}$ (see the attached proof). 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

$\displaystyle \ker T\vert _U = U \cap \ker T, $
where $ T\vert _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

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



"kernel of a linear mapping" is owned by yark. [ full author list (3) | owner history (2) ]
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See Also: linear transformation, image of a linear transformation, nullity, rank-nullity theorem

Other names:  nullspace, null-space, kernel

Attachments:
$\operatorname{ker} L=\{0\}$ if and only if $L$ is injective (Theorem) by Mathprof
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Cross-references: matrices, invertible, finite, injective, function, nullity, vector subspace, vectors, vector spaces, linear mapping
There are 44 references to this entry.

This is version 16 of kernel of a linear mapping, born on 2001-11-13, modified 2007-01-08.
Object id is 807, canonical name is KernelOfALinearTransformation.
Accessed 15699 times total.

Classification:
AMS MSC: 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)
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