- Exercise for the break
Give an example of three vectors in such that each two of them is linearly independent, but all three vectors together are linearly dependent.
- Exercises
Find, for the vectors
-
in , a non-trivial representation of the zero vector.
Find, for the vectors
-
in , a non-trivial representation of the zero vector.
Decide whether the following vectors are
linearly independent.
- , , ,
in the
-vector space
.
- , in the
-vector space
.
- , in the
-vector space
.
- , in the -vector space
.
Show that the three vectors
-
in are
linearly independent.
Let be a
-vector space,
and let be a family of vectors in . Show that the family is
linearly independent
if and only if there exists a
linear subspace
such that the family is a
basis
of .
Determine a
basis
for the
linear subspace
-
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
-
form 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
-
In , let the two
linear subspaces
-
and
-
be given. Determine a basis for .
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
linear subspace.
Show that has a
basis
consisting of vectors, such that all their entries are integer numbers.
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
, .
Let be a
-vector space,
let be a
basis
of , and let
-
be the corresponding bijective mapping in the sense of
Remark 7.12
.
Show that this mapping transforms the componentwise addition in into the vector addition in , that is,
-
holds.
Let be a
basis
of , and let
-
be the corresponding bijective mapping in the sense of
Remark 7.12
.
Show that this mapping is, in general, not compatible with componentwise multiplication in .
Let be a
-vector space,
and let
, ,
be a
basis
of . Let
, ,
be another family of vectors in . Suppose that, for every
,
the equality
-
holds. Show that also
, ,
is a basis of .
Let be the
polynomial ring
over . For
,
set
-
Show that
, ,
is a
basis
of .
Formulate and prove
Theorem 7.11
for an arbitrary
(not necessarily finite)
family of vectors
, .
We consider the real numbers as a
-vector space.
Show that the set of real numbers , where runs through the set of all
prime numbers,
is
linearly independent.
Tip: Use that every positive natural number has a unique representation as a product of prime numbers.
What does
the Theorem of Hamel
mean in the following examples?
- The real numbers as a
-vector space.
- The set of all real seqeunces
-
- The set of all continuous functions from to .
Let be an
ordered field,
and let
-
be the
vector space
of all
sequences
in
(with componentwise addition and scalar multiplication).
a) Show that
(without using theorems about convergent sequences), the set of zero sequences, that is,
-
is a
-linear subspace
of .
b) Are the sequences
-
linearly independent
in ?
- Hand-in-exercises
Establish if in the three vectors
-
form a basis.
Establish if in the two vectors
-
form a basis.
Show that, for the space of all
-matrices
, the matrices that have in position the entry , and elsewhere , 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 an -basis of .
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 the
polynomial ring
over . We set
,
and, for
,
we set
-
Show that
, ,
is a
basis
of .