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student::Kn -- the vectorspace of n-tupels over K

Introduction

The domain student::Kn represents the vectorspace of n-tupels over the field F.

Domain

student::Kn(F)
student::Kn(n,F)

Parameters

F - a field, i.e. a domain of category Cat::Field.
n - a positive integer

Details

Creating Elements

student::Kn(n,F)()
student::Kn(n,F)(listofrows)
student::Kn(n,F)(list)
student::Kn(n,F)(indexfunc)

Parameters

list - list of vector components.
listofrows - list of (at most) n rows. Each row is a list of vector components.
indexfunc - function or functional expression in two parameters (the row and column index).

Categories

Cat::VectorSpace(F)

,Cat::Matrix(F)

Related Domains

Dom::MatrixGroup

Details

Method _mult: multiplies with a scalar

Example 1

Let us create the vector space of the 3-tupel over the field Dom::Real:

>> R3 := student::Kn(3,Dom::Real)
                         student::Kn(3, Dom::Real)

Now we create some elements of this domain in different ways:

>> u := R3([1,2,3]);
   v := R3([[2],[3],[4]]);
   w := R3()
                                  +-   -+
                                  |  1  |
                                  |     |
                                  |  2  |
                                  |     |
                                  |  3  |
                                  +-   -+
      
                                  +-   -+
                                  |  2  |
                                  |     |
                                  |  3  |
                                  |     |
                                  |  4  |
                                  +-   -+
      
                                  +-   -+
                                  |  0  |
                                  |     |
                                  |  0  |
                                  |     |
                                  |  0  |
                                  +-   -+

We perform some calculation with the just created elements. We add the three vectors v, w and u of the vectorspace, multiply the vector w with the scalar 3 and the vector v with the scalar -4:

>> v + w + u;
   3*w;
   v*(-4)
                                  +-   -+
                                  |  3  |
                                  |     |
                                  |  5  |
                                  |     |
                                  |  7  |
                                  +-   -+
      
                                  +-   -+
                                  |  0  |
                                  |     |
                                  |  0  |
                                  |     |
                                  |  0  |
                                  +-   -+
      
                                 +-     -+
                                 |   -8  |
                                 |       |
                                 |  -12  |
                                 |       |
                                 |  -16  |
                                 +-     -+

Example 2

Let us see how we can use a function for creating elements of the domain. The function f computes the square of the given number. So the entry in the i-th row of the constructed vector will be i^2.

>> f := i -> i^2:
   R3 := student::Kn(3,Dom::Real);
   R4 := student::Kn(4,Dom::Real);
   v := R3(f);
   w := R4(f)
                         student::Kn(3, Dom::Real)
      
                         student::Kn(4, Dom::Real)
      
                                  +-   -+
                                  |  1  |
                                  |     |
                                  |  4  |
                                  |     |
                                  |  9  |
                                  +-   -+
      
                                 +-    -+
                                 |   1  |
                                 |      |
                                 |   4  |
                                 |      |
                                 |   9  |
                                 |      |
                                 |  16  |
                                 +-    -+

Example 3

The zero vector is defined by the entry "zero" as we can see in the following example:

>> R3 := student::Kn(3,Dom::Real):
   R3::zero();
   v := R3([[2],[3],[4]]);
   v - R3::zero()
                                  +-   -+
                                  |  0  |
                                  |     |
                                  |  0  |
                                  |     |
                                  |  0  |
                                  +-   -+
      
                                  +-   -+
                                  |  2  |
                                  |     |
                                  |  3  |
                                  |     |
                                  |  4  |
                                  +-   -+
      
                                  +-   -+
                                  |  2  |
                                  |     |
                                  |  3  |
                                  |     |
                                  |  4  |
                                  +-   -+

Super-Domain

Dom::MatrixGroup

Axioms

if F has Ax::canonicalRep
Ax::canonicalRep

Background

Changes




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