# 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 ?

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 ?

• Description: collection of exercices on the definition of vector spaces. This is the main site of WIMS (WWW Interactive Multipurpose Server): interactive exercises, online calculators and plotters, mathematical recreation and games
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