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?

ϰ: vector spaces   linear algebra  

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