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

By Alan G. Hamilton

It is a brief, readable advent to easy linear algebra, as often encountered in a primary path. the improvement of the topic is built-in with numerous labored examples that illustrate the information and strategies. The layout of the booklet, with textual content and proper examples on dealing with pages signifies that the reader can persist with the textual content uninterrupted. the coed may be capable of paintings throughout the booklet and study from it sequentially. pressure is put on functions of the equipment instead of on constructing a logical approach of theorems. quite a few routines are supplied.

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In normal applications of this process there is no need to keep a note of the elementary matrices used. 47 5. 4 illustrates the basis of the procedure. Starting with a square matrix A, the GE process leads to an upper triangular matrix, say A'. In the example, continue as follows. Subtract a multiple of the third row of A' from the second row in order to get 0 in the (2, 3)-place. Next, subtract a multiple of the third row from the first row in order to get 0 in the (1, 3)place. Last, subtract a multiple of the second row from the first row in order to get 0 in the (1, 2)-place.

E,A is an upper triangular matrix, where A= 0 1 -3 1 2 1 1 1 1 4 2 2 We proceed with the standard GE process, noting the elementary matrix which corresponds to each row operation. 1 -3 2 2 1 1 1 1 4 2 1 1 2 1 1 0 1- 3 2 1 1 4 2 f0 1 1 2 1 0 1 -3 2 0 -1 3 l 1 2 1 1 0 1 -3 2 0 0 0 3 I 1 2 1 1 0 1 -3 2 0 0 0 1 ch ange rows i nter E 0 1 0 1 0 0 0 0 1 E 2= l 1 0 0 0 1 1 -1 0 (3) - (1) E 3= (3)+ (2) E 4= (3). 14 0 0 2 1 1 1 -3 2 0 0 1 Elementary matrices and their inverses. 1 10 01 1 01 0 0 has inverse 1 0 0 11 0 0 1 1 0 0 0 1 0 00 5 1 3 0 0 1 0 [ 0 0 1 1 iJ has inverse has inverse 10 0 1 1 0 0 0 1 0 0 0 5 1 -3 0 0 1 0 [0 0 0 The way to see these is to consider the effect of premultiplying by first one and then the other of each given pair.

The case when both p and q are 1 is rather trivial and need not concern us here. A column vector with p entries we shall call a p-vector, so a p-vector is a p x 1 matrix. Addition of matrices (including addition of row or column vectors) is very straightforward. We just add the corresponding entries. 2. The only point to note is that, in order for the sum of two matrices (or vectors) to make sense, they must be of the same size. To put this precisely, they must both be p x q matrices, for the same p and q.

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