- Warm-up-exercises
Compute the following product of matrices
-
Compute, over the complex numbers, the following product of matrices
-
Determine the product of matrices
-
where the
-th standard vector
(of length
)
is considered as a row vector, and the
-th standard vector
(also of length
)
is considered as a column vector.
Let
be a
- matrix. Show that the matrix product
of
with the
-th standard vector
(regarded as column vector),
is the
-th column of
. What is
, where
is the
-th standard vector
(regarded as a row vector)?
Compute the product of matrices
-
according to the two possible parantheses.
For the following statements we will soon give a simple proof using the relationship between matrices and linear maps.
Show that the multiplication of matrices is associative.
More precisely: Let
be a
field,
and let
be an
-matrix,
an
-matrix and
a
-matrix over
. Show that
.
For a matrix
we denote by
the
-th product of
with itself. This is also called the
-th power of the matrix.
Compute, for the matrix
-
![{\displaystyle {}M={\begin{pmatrix}2&4&6\\1&3&5\\0&1&2\end{pmatrix}}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94b8a70b5cd3050f6a16df5e82993cec65183c45)
the powers
-
Let
be a
field,
and let
and
be
vector spaces
over
. Show that the
product set
-
is also a
-vector space.
Let
be a
field,
and
an index set. Show that
-
![{\displaystyle {}K^{I}:=\operatorname {Maps} \,(I,K)\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30056c2fab53591ed6fbed00f2bad9c766bf0f69)
with pointwise addition and scalar multiplication, is a
-vector space.
Let
be a
field
and let
-
be a system of linear equations over
. Show that the set of all solutions of this system is a
linear subspace
of
. How is this solution space related to the solution spaces of the individual equations?
Show that the addition and the scalar multiplication of a
vector space
can be restricted to a
linear subspace,
and that this subspace with the inherited structures of
is a vector space itself.
Let
be a
field,
and let
be a
-vector space.
Let
be subspaces of
. Prove that the union
is a subspace of
if and only if
or
.
- Hand-in-exercises
Compute, over the complex numbers, the following product of matrices
-
We consider the matrix
-
![{\displaystyle {}M={\begin{pmatrix}0&a&b&c\\0&0&d&e\\0&0&0&f\\0&0&0&0\end{pmatrix}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fab8c00b81b8efc07cf6f001386d1329d791689f)
over a field
. Show that the fourth power of
is
, that is
-
![{\displaystyle {}M^{4}=MMMM=0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b4f9090b17152e4abf8f2c6b87234a97589a8fb)
Let
be a
field,
and let
be a
-vector space.
Show that the following properties hold
(for
and
).
- We have
.
- We have
.
- We have
.
- If
and
then
.
Give an example of a vector space
and of three subsets of
which satisfy two of the subspace axioms, but not the third.