!! used as default html header if there is none in the selected theme. OEF Vector space definition

OEF Vector space definition --- Introduction ---

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

Space of maps

Let be the set of maps
,
(i.e., from the set of to the set of ) with rules of addition and multiplication by a scalar as follows: Is with the structure defined above a vector space over ?

Absolute value

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

Affine line

Let be a line in the cartesian plane, defined by an equation , and let be a fixed point on .

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

Is with the structure defined above a vector space over ?

Alternated addition

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

Fields

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

Matrices

Let be the set of real matrices. On , we define the multiplication by a scalar as follows.

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 ?


Matrices II

Is the set of matrices of elements and of , together with the usual addition and scalar multiplication, a vector space over the field of ?

Multiply/divide

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

Non-zero numbers

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

Transaffine

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

Transquare

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

Unit circle

Let be the set of points on the circle in the cartesian plane. For any point in , there is a real number such that , .

We define the addition and multiplication by a scalar on as follows:

Is with the structure defined above a vector space over ?
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