- 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
linear 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
-
and
-
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
-
forms 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 exchange 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 that 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 that 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
that describes the
change of bases
from to .
- Hand-in-exercises
Show that the set of all real
polynomials
of
degree
that 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
-
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 that 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 that 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
that describes the
change of basis
from to .