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**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.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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.

Vector space; Subspaces; Canonical bases

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 . Let be n − 1 linearly independent vectors in the space V where

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)

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

**Ex 2**

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

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

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

**Ex 3**

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

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

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 (a_{1}) – (a_{6}):

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

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)

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

Perform the following operations . We obtain

Consider the elements a_{n,n}, a_{n, 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 a_{n,n} ≠ 0 then perform the operation first, and the operations after the first one. We obtained the matrix

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

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

Perform the operations, . We obtain

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

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

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

Perform the operations . We obtain

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

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

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

Perform the operations . We obtain

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

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

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

Perform the operations

We obtain

At least one element among a_{n, 1}, a_{n, n+1} is not zero in this matrix. If a_{n, 1}≠0 then perform the operation and interchange the rows. We obtain the matrix of the type (1) as it’s needed. If a_{n, 1}=0 then a_{n, n+1}0. Perform the operation first, and the operations 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

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**.

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.

- Holt J (2013) Linear Algebra with Applications, W.H. Freeman and Company, New York.
- Patera J, Sharp RT, Winternitz P, Zassenhaus H (1976) Invariants of Real Low Dimension Lie Algebras. Journal of Mathematical Physics 17: 986-994.
- Patera J, Winternitz P (1977) Subalgebras of real three- and four-dimensional Lie algebras. Journal of Mathematical Physics 18: 1449-1455.
- Bincer AM (2012) Lie Groups and Lie Algebras – A Physicist’s Perspective. Oxford University Press.
- Shtukar U (2016) Canonical Bases for Subspaces of a Vector Space, and 5-dimensional Subalgebras of Lie Algebra of Lorentz Group. Journal of Generalized Lie Theory and Applications 10: 1.

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