- Warm-up-exercises
Let
be a field, and let
be a
-vector space of dimension
.
Suppose that
vectors
in
are given. Prove that the following facts are equivalent.
form a basis for
.
form a system of generators for
.
are linearly independent.
Let
be a field, and let
denote the
polynomial ring
over
. Let
.
Show that the set of all polynomials of degree
is a
finite dimensional
subspace
of
. What is its
dimension?
Show that the set of real
polynomials
of
degree
which have a zero at
and a zero at
is a
finite dimensional
subspace
of
. Determine the
dimension
of this vector space.
Let
be a field, and let
and
be two finite-dimensional
vector spaces with
-
![{\displaystyle {}\dim _{}{\left(V\right)}=n\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab72054f9cfabc8640a4cd323e0bb992ea9a6ceb)
and
-
![{\displaystyle {}\dim _{}{\left(W\right)}=m\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acbfb246ff654aec53181220465fda150380b620)
What is the dimension of the Cartesian product
?
Let
be a finite-dimensional vector space over the complex numbers, and let
be a basis of
. Prove that the family of vectors
-
form a basis for
, considered as a real vector space.
Consider the standard basis
in
and the three vectors
-
Prove that these vectors are linearly independent and extend them to a basis by adding an appropriate standard vector as shown in the
base change theorem.
Can one take any standard vector?
Determine the
transformation matrices
and
,
for the
standard basis
, and the basis
in
, which is given by
-
Determine the
transformation matrices
and
,
for the
standard basis
, and the basis
of
, which is given by the vectors
-
We consider the families of vectors
-
in
.
a) Show that
and
are both a
basis
of
.
b) Let
denote the point which has the coordinates
with respect to the basis
. What are the coordinates of this point with respect to the basis
?
c) Determine the
transformation matrix
which describes the
change of bases
from
to
.
- Hand-in-exercises
Show that the set of all real
polynomials
of
degree
, which have a zero at
, at
and at
, is a
finite dimensional
subspace
of
. Determine the
dimension
of this vector space.
Let
be a field, and let
be a
-vector space. Let
be a family of vectors in
, and let
-
![{\displaystyle {}U=\langle v_{i},\,i=1,\ldots ,m\rangle \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb0f805882675ea82f51f9f1ddf91f57a1a2b63c)
be the
linear subspace
they span. Prove that the family is linearly independent if and only if the dimension of
is exactly
.
Determine the
transformation matrices
and
,
for the
standard basis
, and the basis
of
, which is given by the vectors
-
We consider the families of vectors
-
in
.
a) Show that
and
are both a
basis
of
.
b) Let
denote the point which has the coordinates
with respect to the basis
. What are the coordinates of this point with respect to the basis
?
c) Determine the
transformation matrix
which describes the
change of basis
from
to
.