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?

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 xa.
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?

Other exercises on: vector spaces   linear algebra

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