By Alan G. Hamilton
This can be a brief, readable creation to simple linear algebra, as often encountered in a primary direction. the improvement of the topic is built-in with lots of labored examples that illustrate the guidelines and techniques. The layout of the booklet, with textual content and suitable examples on dealing with pages implies that the reader can persist with the textual content uninterrupted. the coed might be in a position to paintings during the publication and examine from it sequentially. tension is put on functions of the tools instead of on constructing a logical method of theorems. various routines are supplied.
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Additional info for A First Course in Linear Algebra: With Concurrent Examples
E. I ajkbki, k=1 k=1 which is the same as the (i,j)-entry in (AB)T. This proves the result. 9 Examples of skew-symm etric matr ices. 10 -1 -2 1 2 0 -3 3 0 = 2]. 0 -1 --2 1 0 3 2 -3 0 0 T 0 = - -1 -2 1 2 0 -3 3 0 Examples of orthogonal matrices. 1 (i) - [- 02 '=[0 -20] - 2 1 1 T2 T2 Let A = 1 2 11 2_2 1 AT A= 2+2 +2 2-2 /_2 2 1 1 2 so Then AT= 1 T 1 0 0 1 ' 1 2 2 and i0 AAT=L-1+i j. 12 +i 1=10 Hence A is orthogonal. 2 (ii) Let B= 3 - 3 2 3 2 3 -3 1 2 2 Fl' 2 Then BT= 3 2 3 -32 3 3 1 2 3 3 3 -3 1-5 Then by direct evaluation we verify that BTB=1 and BBT =I.
FS=F1F2 ... , F1E1=I. Consequently, AB=F1F2 ... F3ESEs-, ... E2E1=I. Hence B is the inverse of A. Our procedure for finding the inverse of A must therefore calculate for us the product ESES _ 1 ... E2E1. This product can be written as ESES_ 1 ... E2E11, and this gives the hint. We convert A to I by certain elementary row operations. The same row operations convert I into A (if it exists). E11. ) 1 0 0 1 0 0 0 0 1 3 3 (1) - 2 x (3) (2) - 2 x (3) 3 3 3 3 1 1 _ 1 6 3 6 This is the end of the process, since the left-hand matrix is an identity matrix.
Notice that when the GE process is applied to a square matrix the result is always an upper triangular matrix. The main diagonal also plays a part in our next kind of special matrix. A square matrix is symmetric if reflection in the main diagonal leaves the matrix unchanged. e. 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.