Linear Algebra and Geometry is organized around carefully sequenced problems that help students build both the tools and the habits that provide a solid basis for further study in mathematics. Requiring only high school algebra, it uses elementary geometry to build the beautiful edifice of results and methods that make linear algebra such an important field. The materials in Linear Algebra and Geometry have been used, field tested, and refined for over two decades. It is aimed at preservice and practicing high school mathematics teachers and advanced high school students looking for an addition to or replacement for calculus. Secondary teachers will find the emphasis on developing effective habits of mind especially helpful. The book is written in a friendly, approachable voice and contains nearly a thousand problems.
| Acknowledgments |
|
xi | |
| Introduction |
|
xiii | |
|
Chapter 1 Points and Vectors |
|
|
1 | (46) |
|
|
|
3 | (5) |
|
|
|
8 | (11) |
|
|
|
19 | (16) |
|
|
|
35 | (7) |
|
|
|
42 | (1) |
|
|
|
43 | (2) |
|
|
|
45 | (2) |
|
Chapter 2 Vector Geometry |
|
|
47 | (62) |
|
|
|
49 | (2) |
|
|
|
51 | (13) |
|
|
|
64 | (5) |
|
|
|
69 | (7) |
|
|
|
76 | (10) |
|
|
|
86 | (16) |
|
|
|
102 | (1) |
|
|
|
103 | (3) |
|
|
|
106 | (3) |
|
Chapter 3 The Solution of Linear Systems |
|
|
109 | (40) |
|
|
|
111 | (2) |
|
|
|
113 | (12) |
|
|
|
125 | (5) |
|
3.4 Linear Dependence and Independence |
|
|
130 | (6) |
|
3.5 The Kernel of a Matrix |
|
|
136 | (6) |
|
|
|
142 | (1) |
|
|
|
143 | (3) |
|
|
|
146 | (3) |
|
|
|
149 | (60) |
|
|
|
151 | (3) |
|
4.2 Adding and Scaling Matrices |
|
|
154 | (6) |
|
4.3 Different Types of Square Matrices |
|
|
160 | (6) |
|
4.4 Matrix Multiplication |
|
|
166 | (10) |
|
4.5 Operations, Identity, and Inverse |
|
|
176 | (9) |
|
4.6 Applications of Matrix Algebra |
|
|
185 | (14) |
|
|
|
199 | (2) |
|
|
|
201 | (5) |
|
|
|
206 | (3) |
|
Chapter 5 Matrices as Functions |
|
|
209 | (68) |
|
|
|
211 | (2) |
|
5.2 Geometric Transformations |
|
|
213 | (10) |
|
|
|
223 | (13) |
|
5.4 Determinants, Area, and Linear Maps |
|
|
236 | (11) |
|
5.5 Image, Pullback, and Kernel |
|
|
247 | (9) |
|
5.6 The Solution Set for AX = B |
|
|
256 | (8) |
|
|
|
264 | (1) |
|
|
|
265 | (4) |
|
|
|
269 | (2) |
|
|
|
271 | (2) |
|
|
|
273 | (4) |
|
|
|
277 | (62) |
|
|
|
279 | (3) |
|
|
|
282 | (8) |
|
6.3 Representations of Markov Chains |
|
|
290 | (7) |
|
6.4 Applying Matrix Algebra to Markov Chains |
|
|
297 | (8) |
|
6.5 Regular Markov Chains |
|
|
305 | (10) |
|
6.6 Absorbing Markov Chains |
|
|
315 | (14) |
|
6.7 The World Wide Web: Markov and PageRank |
|
|
329 | (7) |
|
|
|
336 | (3) |
|
|
|
339 | (50) |
|
|
|
341 | (2) |
|
7.2 Introduction to Vector Spaces |
|
|
343 | (10) |
|
|
|
353 | (6) |
|
7.4 Linear Span and Generating Systems |
|
|
359 | (9) |
|
7.5 Bases and Coordinate Vectors |
|
|
368 | (14) |
|
|
|
382 | (1) |
|
|
|
383 | (3) |
|
|
|
386 | (3) |
|
Chapter 8 Bases, Linear Mappings, and Matrices |
|
|
389 | (58) |
|
|
|
391 | (2) |
|
|
|
393 | (6) |
|
|
|
399 | (10) |
|
8.4 Building and Representing Linear Maps |
|
|
409 | (13) |
|
|
|
422 | (8) |
|
|
|
430 | (9) |
|
|
|
439 | (1) |
|
|
|
440 | (4) |
|
|
|
444 | (3) |
|
Chapter 9 Determinants and Eigentheory |
|
|
447 | (102) |
|
|
|
449 | (2) |
|
|
|
451 | (15) |
|
9.3 More Properties of Determinants |
|
|
466 | (11) |
|
9.4 Elementary Row Matrices and Determinants |
|
|
477 | (12) |
|
9.5 Determinants as Area and Volume |
|
|
489 | (19) |
|
9.6 Eigenvalues and Eigenvectors |
|
|
508 | (16) |
|
9.7 Topics in Eigentheory |
|
|
524 | (17) |
|
|
|
541 | (1) |
|
|
|
542 | (5) |
|
|
|
547 | (2) |
| Cumulative Review |
|
549 | (3) |
| Cumulative Test |
|
552 | (3) |
| Index |
|
555 | |
Al Cuoco, Education Development Center, Inc., Waltham, MA.
Kevin Waterman, Education Development Center, Inc., Waltham, MA.
Bowen Kerins, Education Development Center, Inc., Waltham, MA.
Elena Kaczorowski, Education Development Center, Inc., Waltham, MA.
Michelle Manes, University of Hawaii, Honolulu, HI.
