Sparse vector space of degree 3 and dimension 1

Question

I'm trying to do some computations involving Lie algebras on SAGE, but I'm not understanding much of the terminology.

When I ask it to describe a certain module over the Lie algebra, it tells me it's a "Sparse vector space of degree 3 and dimension 1 over Rational Field, Basis matrix: [0 1 0]".

a) What is a sparse vector space, as opposed to an ordinary vector space?

b) What is meant by the 'degree' of the vector space here? This is the first I'm hearing about it.

c) How does the matrix it generated relate to the structure of the vector space?

Gustavo Shigueo · Answer 1 · 2023-06-07T21:41:06.633

0

A sparse vector is one that is mostly filled with zeroes.

Dimension 1 means it's a one dimensional (or 1D) array (or a vector), i.e. an array that contains only scalar values (numbers or other data types) and NOT other arrays. Extending that logic a 2D array is an array that contains only other arrays, which in turn contain scalars... Examples:

1D array:
[1, 2, 3]

2D array (usually called a matrix)
[
    [1, 2, 3],
    [4, 5, 6]
]

3D array
[
    [
        [1, 2, 3],
        [1, 2, 3]
    ],
    [
        [1, 2, 3],
        [1, 2, 3]
    ],
    [
        [1, 2, 3],
        [1, 2, 3],
    ]
]

I'm not sure about the degree, but I believe it's is how many scalars the inner most array holds.

As for how a matrix relates to this, matrices are usually represented as 2D arrays, where each inner array is a matrix row

Example of matrix with chess:

[
    ['r', 'b', 'n', 'q', 'k', 'n', 'b', 'r'],
    ['p', 'p', 'p', 'p', 'p', 'p', 'p', 'p'],
    [' ', ' ', ' ', ' ', ' ', ' ', ' ', ' '],
    [' ', ' ', ' ', ' ', ' ', ' ', ' ', ' '],
    [' ', ' ', ' ', ' ', ' ', ' ', ' ', ' '],
    [' ', ' ', ' ', ' ', ' ', ' ', ' ', ' '],
    ['p', 'p', 'p', 'p', 'p', 'p', 'p', 'p'],
    ['r', 'b', 'n', 'q', 'k', 'n', 'b', 'r'],
]

edited Jun 07 '23 at 21:41

answered Jun 07 '23 at 21:34

Gustavo Shigueo

399
3
11

Thanks. So the vector space itself (in my output) is Q. Degree does indeed seem to be the number of scalars the array holds, but again, what does this have to do with a description of the module? If it's trying to give me a basis, [1] alone should suffice. – A.D. Jun 07 '23 at 21:44
I am not at all familiar with Lie Algebra, so I might be unable to help with this part of your question anyway, but I'm also a little confused. What is `the module`? And what is trying to give you a basis, and who (or what) did you ask about thi to get the `Sparse vector...` description? You keep refering to those as `it` and I'm kinda lost – Gustavo Shigueo Jun 07 '23 at 21:48
Nevermind that question, I searched on Google and it seems you're talking about a website called sagemath. Unfortunatelly I still don't know what is a module, so I don't think I can help with the relation between this and `a description of the module` – Gustavo Shigueo Jun 07 '23 at 22:00
Sorry, let me rephrase. Let’s forget about modules. I asked SAGE to describe the structure of a given vector space over a Q. The output was "Sparse vector space..." etc. etc.. If this vector space is of dimension 1 over Q, it of course must be Q itself. On the other hand, I can infer that if the vector space has degree m and dimension n, it’ll be associated with an n*m matrix. What I don’t understand is what this parameter (degree) is supposed to tell us about the vector space. – A.D. Jun 07 '23 at 22:04

John Palmieri · Accepted Answer · 2023-06-07T22:27:58.830

Whether a vector space is "sparse" or not in SageMath determines how its vectors are stored. A vector in a sparse vector space is stored as a dictionary (efficient when most of the entries are zero), where as a vector in a dense vector space might be stored as a tuple. The sparse/dense distinction is explained for matrices at https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/matrix_generic_dense.html and https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/matrix_generic_sparse.html, and vectors work analogously.

In SageMath, the degree of a vector space is the dimension of the ambient space in which it sits. You have a one-dimensional subspace of Q^3, so the degree is three.

The matrix should then be describing how the one-dimensional space is included in Q^3: the rows of the matrix give a basis for the vector space. See the SageMath documentation for free modules for some examples.

Sparse vector space of degree 3 and dimension 1

2 Answers2