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.

If C₁ (resp. C₂) is a circle of center (x₁,y₁) (resp. (x₂,y₂) ) and radius , C₁ + C₂ will be the circle of center (x₁+x₂,y₁+y₂) and radius .
If C is a circle of center (x,y) and radius , and if a is a real number, then aC is a circle of center (ax,ay) and radius .

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:

If f₁ and f₂ are two maps in S, f₁+f₂ is a map f: : -> such that f(x)=f₁(x)+f₂(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 .

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:

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).

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.

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 = .

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:

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) = (ax,ay).

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:

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).

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:

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.

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:

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 (ax(),ay()).

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:

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) = (ax,ay()²).

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:

If (cos(t₁),sin(t₁)) and (cos(t₂),sin(t₂)) are two points in S, their sum is defined to be (cos(t₁+t₂),sin(t₁+t₂)).
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)).

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

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