8/26/2016 Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 1 8/26/2016 Page 274 (1) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Pages 274-275 Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 2 8/26/2016 Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 275 (2) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 3 8/26/2016 Page 276 (1) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 276 (2) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 4 8/26/2016 Page 279a Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 279b Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 5 8/26/2016 Page 280 (1) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 280 (2) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 6 8/26/2016 Page 282 Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 283 (1) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 7 8/26/2016 Page 283 (2) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Anti-symmetric matrix Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 8 8/26/2016 Page 283 (3) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 284 (1) Scalar matrices Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 9 8/26/2016 Page 284 (2) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Case study Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 10 8/26/2016 Page 288 (1) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 288 (2a) Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 11 8/26/2016 Page 288 (2b) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Case study Linear system Electrical network Use Kirchhoff’s current law (KCL) And Kirchhoff’s voltage law (KVL) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 12 8/26/2016 Page 291a Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 291b Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 13 8/26/2016 Page 291c Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 14 8/26/2016 Page 292 (1) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 292 (2) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 15 8/26/2016 Page 292 (3) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 294a Inverse matrix Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 16 8/26/2016 Page 294b Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 294c Silahkan baca buku untuk mendapatkan Interpretasi Geometris dari solusi-solusi tersebut Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 17 8/26/2016 Exactly one solution Gauss elimination Infinitely many solution No solution Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Existence and Uniqueness of Solutions? Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 18 8/26/2016 Page 297 (1) Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 19 8/26/2016 Page 297 (2) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 1 2 3 4 5 6 2 4 6 1 2 3 A 4 5 6 , B1 1 2 3 , B2 4 5 6 , B3 3 3 3 7 8 9 7 8 9 7 8 9 7 8 9 (1) 1 2 3 4 5 6 (1)(2) A (2) 4 5 6 B1 1 2 3 (3) 7 8 9 7 8 9 (1) 1 2 3 2 4 6 2*(1) A (2) 4 5 6 (1) B2 4 5 6 7 8 9 (3) 7 8 9 . (1) 1 2 3 1 2 3 (2)(2)(1) A (2) 4 5 6 B3 3 3 3 (3) 7 8 9 7 8 9 B1, B2 and B3 are all row equivalent to A Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 20 8/26/2016 Page 298 (2) Rank of A is 2 because the first two rows are linearly independent. Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 298 (3) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 21 8/26/2016 Page 299 Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 22 8/26/2016 Page 301 Note: • The dimension of the row space is called the rank of the matrix A • The null space of the matrix A is the set of all n-dimensional column vector x such that A x = 0 Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Theorem 1 Elementary row operations do not change the row space of a matrix Theorem 2 If a matrix A is in row echelon form, then the nonzero rows of A are linearly independent Corollary The rank of a matrix is equal to the number of nonzero rows in its row echelon form Theorem 3 The rank of a matrix A plus the nullity of A equals the number of column of A Row Echelon Form Example nonzero row zero row Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 23 8/26/2016 Pages 302-303a Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Pages 302-303b Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 24 8/26/2016 Page 304 Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 305 (1) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 25 8/26/2016 Page 305 (2) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 305 (3) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 26 8/26/2016 Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Determinant? Solution x1 x1 DETERMINANT a22 b1 a12 b2 a11a22 a21a12 a11b1 a21b2 a11a22 a21a12 Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 27 8/26/2016 Page 306a Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 306b Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 28 8/26/2016 Page 307a Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 307b Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 29 8/26/2016 Page 308a Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 308b Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 30 8/26/2016 Page 309 (1) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 309 (2a) Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 31 8/26/2016 Page 309 (2b) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 310 Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 32 8/26/2016 Pages 310-311a Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Pages 310-311b Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 33 8/26/2016 Page 311 (1) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 311 (2) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 34 8/26/2016 Page 312a Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 312b Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 35 8/26/2016 Page 315 Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 316 Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 36 8/26/2016 Pages 317-318a Basic idea: AX I A I A Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Pages 317-318b Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 37 8/26/2016 Pages 317-318c Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 318 Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 38 8/26/2016 Page 319a Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 319b Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 39 8/26/2016 Page 320 (1) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 320 (2) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 40 8/26/2016 Page 321 (1) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Page 321 (2) Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 41 8/26/2016 Page 322 Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Pages 331-332a Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 42 8/26/2016 Pages 331-332b Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. Pages 331-332c Continued Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 43 8/26/2016 Pages 331-332c Advanced Engineering Mathematics by Erwin Kreyszig Copyright 2007 John Wiley & Sons, Inc. All rights reserved. 44
