- 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 an - matrix. Show that the matrix product of with the -th standard vector
(regarded as a 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 parentheses.
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
-
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
-
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
linear subspaces
of . Prove that the union is a linear subspace of if and only if
or .
- Hand-in-exercises
Compute, over the complex numbers, the following product of matrices
-
We consider the matrix
-
over a field . Show that the fourth power of is , that is,
-
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 that satisfy two of the subspace axioms, but not the third.