Other exercises
Let
be a field and let
and
be
-vector spaces. Let
-
be a linear map. Prove that the graph of the map is a subspace of the Cartesian product
.
Let
be a field and let
be a
-vector space. Prove that for
the map
-
is linear.
How does the graph of a linear map
-
-
-
look like? How can you see in a sketch of the graph the kernel of the map?
Let
be a field and let
and
be
-vector spaces. Let
be a system of generators for
and let
be a family of vectors in
.
a) Prove that there is at most one linear map
-
such that
for all
.
b) Give an example of such a situation, where there is no linear mapping with
for all
.
Let
be a field and let
be a
-matrix and
a
-matrix over
. Prove the following relationships concerning the rank
-
Prove that equality on the left occurs if
is invertible, and equality on the right occurs if
is invertible. Give an example of non-invertible matrices
and
such that equality on the left and on the right occurs.
Prove that the series
-
converges with sum equal to
.
Examine for each of the following subsets
the concepts upper bound, lower bound, supremum, infimum, maximum and minimum.
-
,
-
,
-
,
-
,
-
,
-
,
-
,
-
,
-
.
Explain why the factorial function is continuous.