Due to
exercise,
this is indeed a -vector space.
Let be a
field,
and let
and
be
-
vector spaces. Then the following hold.
- A
linear mapping
-
from another vector space induces a linear mapping
-
- A
linear mapping
-
to another vector space induces a linear mapping
-
Proof
Let be a
field,
and let
and
be
-vector spaces.
Let
-
and
-
de
direct sum decompositions
and let
-
denote the
canonical projections. Then the mapping
-
is an
isomorphism. If we consider as linear subspaces of , then we have the direct sum decomposition
-
It follows directly from
fact
that the given mapping is linear. In order to prove injectivity, let
with
be given. Then there exists some
such that
-
Let
with
.
Then also
for some . Therefore,
for some . Hence
-
In order to prove surjectivity, let a family of homomorphisms , be given, which we consider as mappings to . Then the
-
are linear mappings from to . This yields via
fact
a linear mapping from to , which restricts to the given mappings.
Let denote a
field
and let
and
denote
finite-dimensional
-vector spaces.
Let
be a
basis
of and
be a basis of . Then the assignment
-
is an
isomorphism
of
-vector spaces.
This means that we can consider the direct sum decomposition for the one-dimensional linear subspaces
bzw.
corresponding to bases and apply
fact.
This follows immediately from
fact.