A first course in linear algebra, with concurrent examples by Alan G. Hamilton

By Alan G. Hamilton

This can be a brief, readable creation to simple linear algebra, as often encountered in a primary direction. the advance of the topic is built-in with various labored examples that illustrate the information and techniques. The layout of the publication, with textual content and appropriate examples on dealing with pages implies that the reader can stick with the textual content uninterrupted. the coed may be in a position to paintings during the e-book and study from it sequentially. pressure is put on purposes of the equipment instead of on constructing a logical method of theorems. quite a few routines are supplied.

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Sample text

In fact the arrays which were part of the shorthand way of carrying out the Gaussian elimination process are matrices. The usefulness of matrices originates in precisely that process, but extends far beyond. We shall see in this chapter how the advantages of brevity gained through the use of arrays in Chapters 1 and 2 can be developed, and how out of this development the idea of a matrix begins to stand on its own. An array of numbers with p rows and q columns is called a p x q matrix ('p by q matrix'), and the numbers themselves are called the entries in the matrix.

A product of two symmetric matrices is generally not symmetric. -5 4 -13 -4 which is not symmetric. -5 AB= 4. Special matrices 35 upper triangular matrix. A square matrix which has zeros at all positions above the main diagonal is called a lower triangular matrix. A matrix of one or other of these kinds is called a triangular matrix. Such matrices have convenient properties which make them useful in some applications. 5, that sums and products of upper (lower) triangular matrices are upper (lower) triangular.

Each condition implies the other. 9. In practice it is very useful to use only one condition rather than two. Next, a rule for inverses of products. Suppose that A and B are invertible p x p matrices. Must AB be invertible, and if so what is its inverse? Here is the trick: (AB)(BA 1)=A(BB-')A =AIA-' 1 (rule (v) on page 27) =AA-'=I, and (B-1A-1)(AB)=B-1(A-'A)B =B-'IB=I. ) This rule extends to products of any number of matrices. The order reverses. Indeed, we have come across an example of this already.

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