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

By Alan G. Hamilton

This can be a brief, readable advent to uncomplicated linear algebra, as frequently encountered in a primary path. the improvement of the topic is built-in with a good number of labored examples that illustrate the information and techniques. The structure of the booklet, with textual content and proper examples on dealing with pages signifies that the reader can keep on with the textual content uninterrupted. the coed may be in a position to paintings throughout the publication and study from it sequentially. tension is put on purposes of the equipment instead of on constructing a logical procedure of theorems. various workouts are supplied.

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Sample text

The matrix obtained by reflecting in the main diagonal. A is symmetric if AT = A. Notice that the rows of AT are the columns of A, and vice versa. 6. 7) that sums of symmetric matrices are symmetric, but products in general are not. There are three important rules about transposes. 8 Let A and B be any p x p matrices. Then (AB)T = B T A T . To see this, let the (i,j)-entries of A and B be denoted by aij and bij respectively. The (i, j)entry in (AB)T is the (j, i)-entry in AB, which is P` L ajkbki' k=1 The (i,j)-entry in BTAT is P TT k=1 bikakj, where b is the (i, k)-entry in BT and a k of the transpose, we have definition is the (k, j)-entry in AT.

4 Let (1 2 2 3 1 0 1 1 A= - 3 Find whether A is invertible and, if it is, find A-'. First carry out the standard GE process on A, at the same time performing the same operations on an identity matrix. 1 1 1 1 0 0 1 2 3 0 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 2 0 1 1 0 0 1 1 1 0 0 1 1 -1 1 0 0 1 2 -1 1 0 0 0 -1 1 -1 1 1 0 0 1 0 1 1 1 0 1 2 0 0 1 -1 -1 1 0 0 -1 1 0 0 0 1 1 0 0 0 1 0 0 -1 1 1 (2)-(1) E1= E2= (3)-(2) E3= -1 (3)x -1 1 0 0 0 1 0 0 0 -1 . 10 This is where the standard process ends. The matrix A' referred to in the text is 1 2.

10 is another calculation of an inverse. Rule A square matrix A is invertible if and only if the procedure given above reaches an intermediate stage with matrix A' having is on the main diagonal and Os below it. The definition of the matrix inverse required two conditions: B is the inverse of A if AB = I and BA=I. It can be shown, however, that either one of these conditions is sufficient. Each condition implies the other. 9. In practice it is very useful to use only one condition rather than two.

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