OEF vector space definition --- Introduction ---

This module actually 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 S be the set of all circles on the (cartesian) plane, with rules of addition and multiplication by scalars defined as follows. Is S with the addition and multiplication by scalar defined above is a vector space over the field of real numbers?

Space of maps

Let S be the set of maps

f: ---> ,

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

Is S with the structure defined above is a vector space over R ?

Absolute value

Let S be the set of couples (x,y) of real numbers. We define the addition and multiplication by scalar on S as follows: Is S with the structure defined above is a vector space over R?

Affine line

Let L be a line in the cartesian plane, defined by an equation c1x+c2y=c3, and let =(x,y) be a fixed point on L.

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

Is S with the structure defined above is a vector space over R?

Alternated addition

Let S be the set of couples (x,y) of real numbers. We define the addition and multiplication by scalar on S as follows: Is S with the structure defined above is a vector space over R?

Fields

The set of all , together with the usual addition and multiplication, is it a vector space over the field of ?

Matrices

Let be the set of real matrices. On , we define the multiplicatin by scalar as follows. If is a matrix in , and if is a real number, the multiplication of by the scalar is defined to be the matrix , where .

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


Matrices II

The set of matrices of elements and of , together with the usual addition and multiplication, is it a vector space over the field of ?

Multiply/divide

Let S be the set of couples (x,y) of real numbers. We define the addition and multiplication by scalar on S as follows: Is S with the structure defined above is a vector space over R?

Non-zero numbers

Let S be the set of real numbers. We define addition and multiplication by scalare on S as follows: Is S with the structure defined above is a vector space over R?

Transaffine

Let S be the set of couples (x,y) of real numbers. We define the addition and multiplication by scalar on S as follows: Is S with the structure defined above is a vector space over R?

Transquare

Let S be the set of couples (x,y) of real numbers. We define the addition and multiplication by scalar on S as follows: Is S with the structure defined above is a vector space over R?

Unit circle

Let S be the set of points on the circle x2+y2=1 in the cartesian plane. For any point (x,y) in S, there is a real number t such that x=cos(t), y=sin(t).

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

Is S with the structure defined above is a vector space over R?

Other exercises on: vector spaces   linear algebra  

The most recent version


This page is not in its usual appearance because WIMS is unable to recognize your web browser.

In order to access WIMS services, you need a browser supporting forms. In order to test the browser you are using, please type the word wims here: and press ``Enter''.

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. exercises interactifs, calcul et tracé de graphes en ligne

Keywords: interactive mathematics, interactive math, server side interactivity, algebra, linear algebra, vector space