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Linear Transformations

In: Matrix Algebra From a Statistician’s Perspective

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  • David A. Harville

    (IBM T.J. Watson Research Center, Mathematical Sciences Department)

Abstract

In more advanced presentations of the topics of Chapters 1–21, matrices typically play a subordinate role. In such presentations, the main results are in terms of something called a linear transformation, which is regarded as a more fundamental concept than that of a matrix. The emphasis on linear transformations results in an approach that is more abstract, more “elegant,” and more conducive to geometrical interpretation. It results in more generality in one sense, although not in another sense. It tends to appeal to the “mathematically more sophisticated,” who typically prefer “geometrical reasoning” to “algebraic arguments.” In the first part of the present (and final) chapter of the book, the concept of a linear transformation is formally introduced, and a number of results on linear transformations are derived from first principles (i.e., “independently” of matrices). And, in the latter parts of the chapter, it is shown that linear transformations can be put into 1–1 correspondence with matrices, and then, based on this correspondence, various results and terminology pertaining to matrices are extended to linear transformations.

Suggested Citation

  • David A. Harville, 1997. "Linear Transformations," Springer Books, in: Matrix Algebra From a Statistician’s Perspective, chapter 22, pages 589-620, Springer.
    • Handle: RePEc:spr:sprchp:978-0-387-22677-4_22
    DOI: 10.1007/0-387-22677-X_22
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