See also the collections of exercises on vector spaces in general or definition of subspaces.

- 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 .

,

(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 .

- For any and belonging to , we define .
- For any belonging to and any real number , we define .

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 .

- For any and belonging to , .
- For any belonging to and any real number , .

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 ?

- For any and belonging to , we define .
- For any belonging to and any real number , we define if is non-zero, and .

- 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 .

- 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 .

- For any and belonging to ,
- For any belonging to and any real number , .

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 .

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- Description: collection of exercices on the definition of vector spaces. serveur web interactif avec des cours en ligne, des exercices interactifs en sciences et langues pour l'enseigment primaire, secondaire et universitaire, des calculatrices et traceurs en ligne
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