OEF Vector space definition
--- Introduction ---
This module currently contains 13 exercises on the definition of vector spaces.
Different structures are proposed in each case; up to you to determine whether it is
really a vector space.
See also the collections of exercises on
vector spaces in general or
definition of subspaces.
Circles
Let
be the set of all circles in the (cartesian) plane, with rules of addition and multiplication by scalars defined as follows. - If
(resp.
) is a circle of center
(resp.
) and radius
,
will be the circle of center
and radius
.
- If
is a circle of center
and radius
, and if
is a real number, then
is a circle of center
and radius
.
Is
with the addition and multiplication by a scalar defined above a vector space over the field of real numbers?
Space of maps
Let
be the set of maps
,
(i.e., from the set of to the set of ) with rules of addition and multiplication by a scalar as follows: - If
and
are two maps in
,
is a map
such that
for all
belonging to
.
- If
is a map in
and if
is a real number,
is a map from
to
such that
for all
belonging to
.
Is
with the structure defined above a vector space over
?
Absolute value
Let
be the set of couples
of real numbers. We define the addition and multiplication by a scalar on
as follows: - For any
and
belonging to
, we define
.
- For any
belonging to
and any real number
, we define
.
Is
with the structure defined above a vector space over
?
Affine line
Let
be a line in the cartesian plane, defined by an equation
, and let
be a fixed point on
. We take
to be the set of points on
. On
, we define addition and multiplication by a scalar as follows.
- If
and
are two elements of
, we define
.
- If
is an element of
and if
is a real number, we define
.
Is
with the structure defined above a vector space over
?
Alternated addition
Let
be the set of couples
of real numbers. We define the addition and multiplication by a scalar on
as follows: - For any
and
belonging to
,
.
- For any
belonging to
and any real number
,
.
Is
with the structure defined above a vector space over
?
Fields
Is the set of all , together with the usual addition and multiplication, a vector space over the field of ?
Matrices
Let
be the set of real
matrices. On
, we define the multiplication by a scalar as follows. If
is a matrix in
, and if
is a real number, the product of
by the scalar
is defined to be the matrix
, where
.
Is
together with the usual addition and the above multiplication by a scalar a vector space over
?
Matrices II
Is the set of matrices of elements and of , together with the usual addition and scalar multiplication, a vector space over the field of ?
Multiply/divide
Let
be the set of couples
of real numbers. We define the addition and multiplication by a scalar on
as follows: - For any
and
belonging to
, we define
.
- For any
belonging to
and any real number
, we define
if
is non-zero, and
.
Is
with the structure defined above a vector space over
?
Non-zero numbers
Let
be the set of real numbers. We define addition and multiplication by a scalar on
as follows: - If
and
are two elements of
, the sum of
and
in
is defined to be
.
- If
is an element of
and if
is a real number, the product of
by the scalar
is defined to be
.
Is
with the structure defined above a vector space over
?
Transaffine
Let
be the set of couples
of real numbers. We define the addition and multiplication by a scalar on
as follows: - If
and
are two elements of
, their sum in
is defined to be the couple
.
- If
is an element of
, and if
is a real number, the product of
by the scalar
in
is defined to be the couple
.
Is
with the structure defined above a vector space over
?
Transquare
Let
be the set of couples
of real numbers. We define the addition and multiplication by a scalar on
as follows: - For any
and
belonging to
,
- For any
belonging to
and any real number
,
.
Is
with the structure defined above a vector space over
?
Unit circle
Let
be the set of points on the circle
in the cartesian plane. For any point
in
, there is a real number
such that
,
. We define the addition and multiplication by a scalar on
as follows:
- If
and
are two points in
, their sum is defined to be
.
- If
is a point in
and if
is a real number, the product of
by the scalar
is defined to be
.
Is
with the structure defined above a vector space over
?
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