Linear mapping/Linear subspaces/Kernel/Introduction/Section
A typical property of a linear mapping is that it maps lines to lines (or to a point). More general is following statement.
Let be a field, let and denote -vector spaces and let
be a
-linear mapping. Then the following hold.- For a linear subspace , the image is a linear subspace of .
- In particular, the image of the mapping is a linear subspace of .
- For a linear subspace , the preimage is a linear subspace of .
- In particular, is a linear subspace of .
Proof
Let denote a field, let and denote -vector spaces, and let
denote a -linear mapping. Then
Due to the statement above, the kernel is a linear subspace of .
For an -matrix , the kernel of the linear mapping
given by is just the solution space of the homogeneous linear system
The following criterion for injectivity is important.
Let denote a field, let and denote -vector spaces, and let
denote a -linear mapping. Then is injective if and only if
holds.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
.