Linear mapping/Kernel/Injectivity criterion/Section
Definition
Let denote a field, let and denote -vector spaces, and let
denote a -linear mapping. Then
The kernel is a linear subspace of .
The following criterion for injectivity is important.
Lemma
Let denote a field, let and denote -vector spaces, and let
denote a -linear mapping. Then is injective if and only if holds.
Proof
If the mapping is injective, then there can exist, apart from
,
no other vector
with
.
Hence,
.
So suppose that
,
and let
be given with
.
Then, due to linearity,
Therefore,
,
and so
.