#
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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- Description: collection of exercices on the definition of vector spaces. Serveur Wims de l'ESPE-Nice-Toulon - Université de Nice - Sophia Antipolis
- Keywords: interactive mathematics, interactive math, server side interactivity, algebra, linear_algebra, vector_space