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Classification of Canonical Bases for (<em>n-2</em>)-Dimensional Subspaces of n-Dimensional Vector Space | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Classification of Canonical Bases for (n-2)-Dimensional Subspaces of n-Dimensional Vector Space

Uladzimir Shtukar*

Associate Professor, Math/Physics Department, North Carolina Central University, Durham

*Corresponding Author:
Uladzimir Shtukar
Associate Professor, Math/Physics Department
North Carolina Central University
1801 Fayetteville Street, Durham
NC 27707, USA
Tel: 919-597-0375
E-mail: [email protected]

Received Date: August 25, 2016; Accepted Date: September 21, 2016; Published Date: September 30, 2016

Citation: Shtukar U (2016) Classification of Canonical Bases for (n−1)−dimensional Subspaces of n−Dimensional Vector Space. J Generalized Lie Theory Appl 10:241. doi:10.4172/1736-4337.1000241

Copyright: © 2016 Shtukar U. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

Canonical bases for (n-1)-dimensional subspaces of n-dimensional vector space are introduced and classified in the article. This result is very prospective to utilize canonical bases at all applications. For example, maximal subalgebras of Lie algebras can be found using them.

Keywords

Vector space; Subspaces; Canonical bases

Introduction

The canonical bases for (n-1)-dimensional subspaces of n− dimensional vector space are introduced in the article, and all nonequivalent of them are classified (Theorem 2). This result generalizes a particular result for 5-dimensional subspaces of 6-dimensional vector space obtained in the previous article of the same author. To analyze the general case, reduced row echelon forms of matrices are utilized; about reduced row echelon forms [1]. In addition to the principal result, all nonequivalent reduced row echelon forms for (n−1)×n matrices of the rank (n−1) are found and listed (Theorem 1).

Let V be an n− dimensional vector space with its standard basis image. Let image be n − 1 linearly independent vectors in the space V where

image

The vectors (I) describe the possible bases for any V -dimensional subspace S of V.

Definition 1

Two bases are called equivalent if they generate the same subspace of V, and they are called nonequivalent if they generate two different subspaces of V [2].

We will associate the following (n−1) ×n matrix M with a basis (I)

image

Definition 2

Two matrices are called row equivalent (or just equivalent) if they have the same reduced row echelon form, and they are called nonequivalent if they have different reduced row echelon forms.

Definition 3

The basis (I) is called canonical if its vectors image are the corresponding rows in some reduced row echelon form of the matrix M.

Thus, there is one-to-one correspondence between nonequivalent canonical bases for (n−1)-dimensional subspaces of n−dimensional vector space and reduced row echelon forms for (n−1) ×n matrix M.

Remark

The standard linear operations with rows (vectors) will be utilized: (a) interchange any two rows, (b) multiply any row by a nonzero constant, (c) add a multiple of some row to another row [3].

Consider some examples with nonequivalent canonical bases for subspaces of small dimensional vector spaces.

Ex 1

Let V be 2-dimensional vector space with its standard basis image Each 1-dimensional subspace S of V can be described as image where image. At least one component among a11, a12 of the vector image is not zero. If a12≠0 then perform the operation image, and we obtain the first canonical basis image. If a12=0 then perform the similar operation image,and we obtain the second canonical basis image. These two canonical bases are nonequivalent [4].

Ex 2

Let V be 3-dimensional vector space with its standard basis image. Consider any 2-dimensional subspace S of V that can be described as image where

image

This basis is equivalent to one and only one canonical basis from the next list

image

Details of evaluation are omitted because it is easy. These last canonical bases generate the following matrices associated with them

image

Ex 3

Let V be 6-dimensional vector space with its standard basis image Consider any 5-dimensional subspace S of V that can be described as image where

image

These bases can be transformed into one and only one canonical basis from the next list

image

All these canonical bases are nonequivalent. This result is obtained by the direct evaluation that generalizes Gauss-Jordan elimination method. The necessary details can be found in the different article of the same author [5].

The following matrices are associated with the last canonical bases (a1) – (a6):

image

The matrices obtained in Examples 2 and 3 are particular cases of the following matrices described by the next statement.

Theorem 1

All nonequivalent reduced row echelon forms of (n−1)×n matrices (n 3) of the rank (n−1) are

image

image

We use the mathematical induction method. The statement is true for n=3 and n=6 according Examples 2 and 3.

Suppose that the statement is true for some dimension n, and prove it for the next dimension n + 3. For it, consider the following matrix of the size n ×(n+1)

image

Go to the (n−1) ×n submatrix located in the left upper corner of M′. According the assumption, this submatrix can be transformed into one and only one reduced row echelon form described in the cases (1), (2), (3), (4), ….(n−1), (n). This means that we can replace the mentioned submatrix by one of the given reduced forms, and analyze the new matrix. We will analyze and show all details in steps 1, 2, 3, and steps (n-1), (n). All other steps are very similar to the steps 1, 2, 3, therefore they are omitted.

Step 1

Substitute the left upper (n−1) ×n submatrix in M by the matrix (1). We have the following matrix

image

Perform the following operations image. We obtain

image

Consider the elements an,n, an, n+1 in the last matrix. At least one of them is not zero because the rank of the matrix M′ is equal n. If an,n ≠ 0 then perform the operation image first, and the operations image after the first one. We obtained the matrix

image

This is the matrix of the type (1) as it’s needed. If an, n = 0 then an,n+1 ≠ 0. Perform the operation image first, and the operations image We obtain the matrix

image

This is the matrix of the type (2) as it’s needed. Step 1 is done.

Step 2

Substitute the left upper (n−1) ×n submatrix in M by the matrix (2). We have

image

Perform the operations, image. We obtain

image

At least one element among an, n1, an, n+1 is not zero. If an, n1≠0 then perform the operation image first, and the operations image after the first operation. We obtain the following matrix

image

The last matrix is row equivalent to the matrix (1) as it’s needed. If an, n−1=0 then an, n+1≠ 0. Perform the operation image first, and the operations image after the first one. We obtain the following matrix

image

The last matrix is a matrix of the type (3) as it’s needed.

Step 3

Substitute the left upper (n−1) ×n submatrix in M by the matrix (3). We have

image

Perform the operations image. We obtain

image

At least one element among an, n2, an, n+1 is not zero in this matrix. If an, n2 then perform the operation image first, and the operations image after the first operation. We obtain the following matrix

image

The last matrix is row equivalent to the matrix (1) as it’s needed. If an, n−2= 0 then an, an,n+1≠0. Perform the operation, image first, and the operations image after the first one. We obtain the following matrix

image

We have the matrix that is equivalent to the matrix of the type (4) as it’s needed.

Step (n−1). Substitute the left upper (n−1) ×n submatrix in M′ by the matrix (n−1). We have

image

Perform the operations image. We obtain

image

At least one element among an, 2, an, n+1 is not zero in this matrix. If an, 2≠0 then perform the operation image first, and the operation image after the first operation. We obtain the following matrix

image

If interchange rows in the last matrix, we obtain the matrix (1) as it’s needed. If an, 2=0 then an, n+1≠0. Perform the operation image first, and the operations image after the first one. We obtain the following matrix

image

It is the matrix of the type (n-1) as we need.

Step n

Substitute the left upper (n−1) ×n submatrix in M by the matrix (n). We have

image

Perform the operations image

We obtain

image

At least one element among an, 1, an, n+1 is not zero in this matrix. If an, 1≠0 then perform the operation image and interchange the rows. We obtain the matrix of the type (1) as it’s needed. If an, 1=0 then an, n+10. Perform the operation image first, and the operations image after the first one. We obviously obtain the matrix of the type (n) as it’s needed. The proof is done.

Theorem 1 can be transformed into the next statement that describes canonical bases for (n−1) −dimensional subspaces of n− dimensional vector space.

Theorem 2

All nonequivalent canonical bases for (n−1) −dimensional subspaces of n−dimensional vector space (if (n≥2)) are

image

The statement is true for n≥3 because of Theorem 1. The additional case n = 2 is included in Theorem 2 because of Example 1.

Final Remark

The canonical bases introduced in the article are a powerful instrument that can be utilized in all applications of Linear Algebra. For example, all maximal subalgebras of any Lie algebra can be found using canonical bases listed in Theorem 2.

References

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