- Exercises
Write in the vector
-
as a linear combination of the vectors
-
Write in the vector
-
as a
linear combination
of the vectors
-
Let be a
field,
and let be a
-vector space.
Show that the following statements hold.
- For a family
, ,
of elements in ,
linear span
is a
linear subspace
of .
- The family
, ,
is a spanning system of if and only if
-
Let be a
field,
and let be a
-vector space.
Let
, ,
be a family of vectors in and
, ,
another family of vectors in . Then, for the
spanned linear subspaces,
the inclusion
holds, if and only if
holds for all
.
Let be a
field,
and let be a
-vector space.
Let ,
,
be a family of vectors in , and let
be another vector. Assume that the family
-
is a system of generators of , and that is a linear combination of the ,
.
Prove that also ,
,
is a system of generators of .
We consider in the
linear subspaces
-
and
-
Show that
.
Show that the three vectors
-
in are
linearly independent.
Find, for the vectors
-
in , a non-trivial representation of the zero-vector.
Give an example of three vectors in such that each two of them is linearly independent, but all three vectors together are linearly dependent.
Let be a field, let be a -vector space and let
, ,
be a family of vectors in . Prove the following facts.
- If the family is linearly independent, then for each subset
,
also the family
,
is linearly independent.
- The empty family is linearly independent.
- If the family contains the null vector, then it is not linearly independent.
- If a vector appears several times in the family, then the family is not linearly independent.
- A vector is linearly independent if and only if
.
- Two vectors
and
are linearly independent if and only if is not a scalar multiple of and vice versa.
Let be a field, let be a -vector space, and let
,
be a family of vectors in . Let
,
be a family of elements in . Prove that the family
, ,
is linearly independent
(a system of generators of , a basis of ),
if and only if the same holds for the family
, .
Determine a basis for the solution space of the linear equation
-
Determine a basis for the solution space of the linear system of equations
-
Prove that in , the three vectors
-
are a basis.
Establish if in the two vectors
-
form a basis.
Let be a field. Find a linear system of equations in three variables, whose solution space is exactly
-
Let be a field, and let
-
be a nonzero vector. Find a linear system of equations in variables with equations, whose solution space is exactly
-
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 all real
polynomials
of
degree
, which have a zero for and for , form a
finite-dimensional
linear subspace
in . Determine its
dimension.
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
-
form a basis for , considered as a real vector space.
Let be a
finite field
with elements, and let be an
-dimensional
vector space.
Let be an enumeration
(without repetitions)
of the elements from . After how many elements can we be sure that these form a
generating system
of .
- Hand-in-exercises
Write in the vector
-
as a linear combination of the vectors
-
Prove that it cannot be expressed as a linear combination of two of the three vectors.
We consider in the
linear subspaces
-
and
-
Show that
.
Establish if in the three vectors
-
form a basis.
Establish if in the two vectors
-
form a basis.
Let be the -dimensional standard vector space over , and let
be a family of vectors. Prove that this family is a -basis of if and only if the same family, considered as a family in , is a -basis of .
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
-
be the
linear subspace
they span. Prove that the family is linearly independent if and only if the dimension of is exactly .