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 .

Please take note that WIMS pages are interactively generated; they are not ordinary HTML files. They must be used interactively ONLINE. It is useless for you to gather them through a robot program.

- 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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