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

- If C
_{1}(resp. C_{2}) is a circle of center (x_{1},y_{1}) (resp. (x_{2},y_{2}) ) and radius , C_{1}+ C_{2}will be the circle of center (x_{1}+x_{2},y_{1}+y_{2}) and radius . - If C is a circle of center (x,y) and radius , and if
*a*is a real number, then*a*C is a circle of center (*a*x,*a*y) and radius .

(i.e., from the set of to the set of ) with rules of addition and multiplication by scalar as follows:

- If
*f*_{1}and*f*_{2}are two maps in S,*f*_{1}+*f*_{2}is a map*f*: : -> such that*f*(x)=*f*_{1}(x)+*f*_{2}(x) for all x belonging to . - If
*f*is a map in S and if*a*is a real number,*af*is a map from to such that (*af*)(x)=*a*·*f*(x) for all x belonging to .

- For any (x,y) and (x,y) belonging to
*S*, we define (x,y)+(x,y) = (x+x,y+y). - For any (x,y) belonging to
*S*and any real number*a*, we define*a*(x,y) = (|*a*|x,|*a*|y).

We take *S* to be the set of points on *L*. On *S*, we define addition and multiplication by scalar as follows.

- If =(x,y) and =(x,y) are two elements of
*S*, we define + = . - If =(x,y) is an element of
*S*and if is a real number, we define = .

- For any (x,y) and (x,y) belonging to
*S*, (x,y)+(x,y) = (x+y,y+x). - For any (x,y) belonging to
*S*and any real number*a*,*a*(x,y) = (*a*x,*a*y).

Is together with the usual addition and the above multiplication by scalar a vector space over ?

- For any (x,y) and (x,y) belonging to
*S*, we define (x,y)+(x,y) = (x+x,y+y). - For any (x,y) belonging to
*S*and any real number*a*, we define*a*(x,y) = (x/*a*, y/*a*) if*a*is non-zero, and 0(x,y)=(0,0).

- If
*x*and*y*are two elements of*S*, the sum of*x*and*y*in*S*is defined to be*xy*. - If
*x*is an element of*S*and if*a*is a real number, the multiplication of*x*by the scalare*a*is defined to be*x*.^{a}

- If (x,y) and (x,y) are two elements of
*S*, their sum in*S*is defined to be the couple (x+x,y+y). - If (x,y) is an element of
*S*, and if*a*is a real number, the multiplication of (x,y) by the scalar*a*in*S*is defined to be the couple (*a*x(),*a*y()).

- For any (x,y) and (x,y) belonging to
*S*, (x,y)+(x,y) = (x+x,y+y). - For any (x,y) belonging to
*S*and any real number*a*,*a*(x,y) = (*a*x,*a*y()^{2}).

We define the addition and multiplication by scalare on *S* as follows:

- If (cos(
*t*_{1}),sin(*t*_{1})) and (cos(*t*_{2}),sin(*t*_{2})) are two points in*S*, their sum is defined to be (cos(*t*_{1}+*t*_{2}),sin(*t*_{1}+*t*_{2})). - If
*p*=(cos(*t*), sin(*t*)) is a point in*S*and if*a*is a real number, the multiplication of*p*by the scalar*a*is defined to be (cos(*at*), sin(*at*)).

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- Description: collection of exercices on the definition of vector spaces. interactive exercises, online calculators and plotters, mathematical recreation and games
- Keywords: interactive mathematics, interactive math, server side interactivity, algebra, linear algebra, vector space