LINEAR ALGEBRA and VECTOR GEOMETRY Volume 2 of 2 September 2014 edition Because the book is so large, the entire Linear Algebra course has been split into two olumes. Grant Skene for Grant s Tutoring (www.grantstutoring.com) DO NOT RECOPY Grant s Tutoring is a priate tutoring organization and is in no way affiliated with the Uniersity of Manitoba.
While studying this book, why not hear Grant explain it to you? Contact Grant for info about purchasing Grant s Audio Lectures. Some concepts make better sense when you hear them explained. Better still, see Grant explain the key concepts in person. Sign up for Grant s Weekly Tutoring or attend Grant s Exam Prep Seminars. Text or Grant (204) 489-2884 or go to www.grantstutoring.com to find out more about all of Grant s serices. Seminar Dates will be finalized no later than Sep. 25 for first term and Jan. 25 for second term. HOW TO USE THIS BOOK I hae broken the course up into lessons. Study each lesson until you can do all of my lecture problems from start to finish without any help. Then do the Practise Problems for that lesson. If you are able to sole all the Practise Problems I hae gien you, then you should hae nothing to fear about your Midterm or Final Exam. I hae presented the course in what I consider to be the most logical order. Although my books are designed to follow the course syllabus, it is possible your prof will teach the course in a different order or omit a topic. It is also possible he/she will introduce a topic I do not coer. Make sure you are attending your class regularly! Stay current with the material, and be aware of what topics are on your exam. Neer forget, it is your prof that decides what will be on the exam, so pay attention. Note that the Distance Ed course does Lesson 9 and Lesson 10 first in my book. It then goes back to Lesson 1 and follows sequentially from there. If you hae any questions or difficulties while studying this book, or if you beliee you hae found a mistake, do not hesitate to contact me. My phone number and website are noted at the bottom of eery page in this book. Grant s Tutoring is also in the phone book. I welcome your input and questions. Wishing you much success, Grant Skene Owner of Grant s Tutoring Grant Skene for Grant s Tutoring (text or call (204) 489-2884) DO NOT RECOPY Grant s Tutoring is a priate tutoring organization and is in no way affiliated with the Uniersity of Manitoba.
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TABLE OF CONTENTS FOR VOLUME 1 (These Lessons are in Volume 1) Lesson 1: Systems of Linear Equations The Lecture... 1 The Lecture Problems... 19 Lesson 2: Row-Reduction and Linear Systems Lecture Problems... 20 The Lecture... 24 Homework and Practise Problems... 72 Solutions to Practise Problems... 80 Lesson 3: Matrix Math Important Matrix Facts and Definitions.... 85 Lecture Problems... 87 The Lecture... 88 Homework and Practise Problems... 113 Solutions to Practise Problems... 119 Lesson 4: The Inerse of a Matrix and Applications Lecture Problems... 126 The Lecture... 128 Homework and Practise Problems... 137 Solutions to Practise Problems... 150 Lesson 5: Elementary Matrices Lecture Problems... 158 The Lecture... 160 Homework and Practise Problems... 183 Solutions to Practise Problems... 185 Lesson 6: Determinants and Their Properties Important Determinant Facts and Properties.... 187 Lecture Problems... 189 The Lecture... 191 Homework and Practise Problems... 221 Solutions to Practise Problems... 231 Lesson 7: The Adjoint Matrix Lecture Problems... 236 The Lecture... 237 Homework and Practise Problems... 254 Solutions to Practise Problems... 260 Lesson 8: Cramer s Rule Lecture Problems... 264 The Lecture... 265 Homework and Practise Problems... 267 Solutions to Practise Problems... 271 THE MIDTERM EXAM NORMALLY COVERS LESSONS 1 TO 8. Grant Skene for Grant s Tutoring (www.grantstutoring.com) DO NOT RECOPY
TABLE OF CONTENTS FOR VOLUME 2 (These Lessons are in Volume 2) Lesson 9: Vectors Important Vector Formulas and Facts... 273 Lecture Problems... 275 The Lecture... 276 Homework and Practise Problems... 313 Solutions to Practise Problems... 319 Lesson 10: Lines and Planes Equations of Lines and Planes, etc.... 325 Lecture Problems... 326 The Lecture... 327 Homework and Practise Problems... 346 Solutions to Practise Problems... 352 Lesson 11: Vector Spaces and Subspaces The Definition of Subspace and The 10 Axioms of Vector Space.... 358 Lecture Problems... 359 The Lecture... 361 Homework and Practise Problems... 391 Solutions to Practise Problems... 394 Lesson 12: Linear Independence Key Definitions and Facts.... 398 Lecture Problems... 399 The Lecture... 400 Homework and Practise Problems... 424 Solutions to Practise Problems... 428 Lesson 13: Basis and Dimension The Definition of Basis.... 433 Lecture Problems... 434 The Lecture... 436 Homework and Practise Problems... 450 Solutions to Practise Problems... 459 Lesson 14: Marko Analysis Important Facts about Marko Analysis... 467 Lecture Problems... 468 The Lecture... 470 Homework and Practise Problems... 500 Solutions to Practise Problems... 503 Lesson 15: Linear Transformations Important Facts and Defintions about Linear Transformations... 15-1 Lecture Problems... 15-10 The Lecture... 15-13 Lesson 16: Eigenalues & Eigenectors Important Facts and Defintions about Eigenalues & Eigenectors... 16-1 Lecture Problems... 16-2 The Lecture... 16-3 THE MIDTERM EXAM NORMALLY COVERS LESSONS 1 TO 8. Grant Skene for Grant s Tutoring (phone (204) 489-2884) DO NOT RECOPY
(Linear Algebra) LESSON 9: VECTORS 273 Lesson 9: Vectors Important Vector Formulas and Facts: } If is the ector (a, b, c), which could also be denoted ai + bj + ck, then the length of (also called the norm of ) is denoted, where: } Similarly, if = (a, b, c, d), then ectors of any size, R n. a b c 2 2 2 a b c d 2 2 2 2. The pattern holds for } A unit ector is a ector whose length is exactly 1 unit (i.e., the norm of the ector is 1). Three standard unit ectors are i, the unit ector in the positie x direction, j, the unit ector in the positie y direction, and k, the unit ector in the positie z direction. } The unit ector in the direction of any ector can be denoted ˆ, where: ˆ 1 } Gien two ectors, u and, where the dot product u 0, then u is orthogonal to (u and make a right angle). } The cosine of the angle,, between two ectors, u and, is gien by: cos u u } If u is positie, then 0 90. Which is to say, the angle,, between the two ectors, u and, is an acute angle. } If u is negatie, then 90 180. Which is to say, the angle,, between the two ectors, u and, is an obtuse angle. SAMPLE 1997-2011 Grant Skene for Grant s Tutoring (www.grantstutoring.com) DO NOT RECOPY
274 LESSON 9: VECTORS (Linear Algebra) } Gien two ectors, u and, the orthogonal projection of u onto is denoted proj u, where: proj u u 2 } Gien two ectors, u and, the cross product, u, produces a third ector, w, which is orthogonal to both u and. In fact, w is normal to the entire plane containing u and (i.e., w makes a right angle with the u, plane). } u u but u u; howeer, u u. } The sine of the angle,, between two ectors, u and, is gien by: sin u u } The area of a parallelogram with adjacent sides defined by the ectors u and is gien by: Area of a parallelogram = u } The area of a triangle with two sides defined by the ectors u and is gien by: Area of a triangle = 1 2 u } The olume of a parallelepiped with adjacent edges defined by the ectors u, and w is gien by: Volume of a parallelepiped = u w SAMPLE (Actually, you can pick any pair you want to compute the cross product with, then compute the dot product with the remaining ector. The important thing is do the cross product first, then the dot product.) Volume of the parallelepiped = u w also, or Volume of the parallelepiped = w u also. 1997-2011 Grant Skene for Grant s Tutoring (text or call (204) 489-2884) DO NOT RECOPY
(Linear Algebra) LESSON 9: VECTORS 275 Lecture Problems: (Each of the questions below will be discussed and soled in the lecture that follows.) 1. Let u=(3, 1, 4) and =( 2, 1, 3). Find the following: (a) u (b) u (c) u (d) 2u 3 (e) The cosine of the angle between u and. (f) A unit ector in the direction of u. (g) The projection of u onto. (h) A ector, w, which is orthogonal to u and. (i) A ector, m, which is orthogonal to u but not orthogonal to. (j) The area of the triangle with adjacent sides u and. 2. Let =(2, 7, 1). Find a ector that is 10 units long, but pointing in the exact opposite direction to. 3. (a) Let 1 = ( 1, 2, 2, 4), 2 = (3, 1, 1, 3), and 3 = (p, 3, q, 4). Find p and q such that 3 is orthogonal to both 1 and 2. (b) Let u = (1, 2, 3, 4) and = (2, 5, 6, 7). Find a non-zero R 4 ector w that is orthogonal to both u and. 4. Gien the three points A =(2, 5, 2), B = (1, 2, 3), and C = ( 6, 1, 4). (a) Compute AB and CB. (b) If ABCD is a parallelogram, find the coordinates of point D. 5. Let the points A = (0, 0), B = (4, 0), C = (6, 6), and D = (2, 6) be the ertices of parallelogram ABCD. Use ectors to find the area of that parallelogram. 6. ABCD is a parallelogram. Let u = AD and = AB. (a) Write the ector AC as a linear combination of the ectors u and. (b) Write the ector BD as a linear combination of the ectors u and. A B (c) Use ector methods to show the diagonals of a parallelogram (lines AC and BD) intersect at right angles if and only if the parallelogram is a rhombus. (A rhombus is a parallelogram where all 4 sides are of equal length.) 7. Use ectors to find the distance between the point P = (4, 5) and the line 2x + 3y = 6. SAMPLE D C 1997-2011 Grant Skene for Grant s Tutoring (www.grantstutoring.com) DO NOT RECOPY
276 LESSON 9: VECTORS (Linear Algebra) SAMPLE 1997-2011 Grant Skene for Grant s Tutoring (text or call (204) 489-2884) DO NOT RECOPY