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An introduction to Linear Algebra

Linear algebra is the language of chemometrics. One cannot expect to truly understand most chemometric techniques without a basic understanding of linear algebra. This article reviews the basics of linear algebra and provides the reader with the foundation required for understanding most chemometrics literature. It is presented in a rather dense fashion: no proofs are given and there is little discussion of the theoretical implications of the theorems and results presented. The goal has been to condense into as few pages as possible the aspects of linear algebra used in most chemometric methods. Readers who are somewhat familiar with linear algebra may find this article to be. | An Introduction to Linear Algebra Barry M. Wise and Neal B. Gallagher Eigenvector Research Inc. 830 Wapato Lake Road Manson WA 98831 USA bmw@eigenvector.com Linear algebra is the language of chemometrics. One cannot expect to truly understand most chemometric techniques without a basic understanding of linear algebra. This article reviews the basics of linear algebra and provides the reader with the foundation required for understanding most chemometrics literature. It is presented in a rather dense fashion no proofs are given and there is little discussion of the theoretical implications of the theorems and results presented. The goal has been to condense into as few pages as possible the aspects of linear algebra used in most chemometric methods. Readers who are somewhat familiar with linear algebra may find this article to be a good quick review. Those totally unfamiliar with linear algebra should consider spending some time with a linear algebra text. In particular those by Gilbert Strang are particularly easy to read and understand. Several of the numerical examples in this section are adapted from Strang s Linear Algebra and Its Applications Second Edition Academic Press 1980 . MATLAB The MathWorks Inc. Natick MA commands for performing the operations listed are also included the reader is encouraged to run the examples presented in the text. Those unfamiliar with MATLAB may wish to read the first few sections of the tutorial chapter of the MATLAB User s Guide. Scalars Vectors and Matrices A scalar is a mathematical quantity that is completely described by a magnitude i.e. a single number. Scalar variables are generally denoted by lowercase letters e.g. a. Examples of scalar variables include temperature density pressure and flow. In MATLAB a value can be assigned to a scalar at the command line e.g. a 5 Here we have used the semicolon operator to suppress the echo of the result. Without this semicolon MATLAB would display the result of the assignment a 5 a 5