IJPAM: Volume 79, No. 4 (2012)

INJECTIVITY AND SURJECTIVITY EQUIVALENCY
OF LINEAR MAPPING: A REVERSAL OF
THE TRADITIONAL APPROACH

H. Arsham
University of Baltimore
Baltimore, MD 21201, USA


Abstract. Equivalency of injectivity and surjectivity of linear mapping on R$^{\mathrm{n}}$ is determined by an explicit formulation for pullback of any vector by Gauss-Jordan row (GJR) operations, not by matrix inversion that is the traditional approach.

It is shown that the necessary and sufficient conditions for the equivalency are iff the complete GJR operations can be performed successfully on matrix transformation.

The inverse matrix that is needed for the proof of the equivalency in the standard approach falls out as a by-product.

The nice property of this symbolic approach is considerable columns reduction with savings in both computer time and storage. Finite computational complexities are given.

This reversal to the equivalency enables us to use the system of equations module rather than matrix inversion module in using symbolic computation systems, such as MAPLE.

Received: May 23, 2012

AMS Subject Classification: 08B30, 15A09, 33F10, 47L05

Key Words and Phrases: injectivity and surjectivity equivalency, symbolic matrix inversion, Gauss-Jordan matrix inversion is not operation, symbolic, Cramer rule, Matrix inversion by solving system of equations, symbolic computation systems

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 79
Issue: 4