LINEAR ALGEBRA. and VECTOR GEOMETRY Volume 2 of 2 September 2014 edition

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1 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 ( DO NOT RECOPY Grant s Tutoring is a priate tutoring organization and is in no way affiliated with the Uniersity of Manitoba.

2 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) or go to 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) ) 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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4 Four ways Grant can help you: Grant s Study Books Basic Statistics 1 (Stat 1000) Basic Statistics 2 (Stat 2000) Linear Algebra and Vector Geometry (Math 1300) Matrices for Management (Math 1310) Intro Calculus (Math 1500 or Math 1510) Calculus for Management (Math 1520) Calculus 2 (Math 1700 or 1710) All these books are aailable at UMSU Digital Copy Centre, room 118 Uniersity Centre, Uniersity of Manitoba. Grant s books can be purchased there all year round. You can also order a book from Grant directly. Please allow one business day because the books are made-to-order. Grant s One-Day Exam Prep Seminars These are one-day, 12-hour marathons designed to explain and reiew all the key concepts in preparation for an upcoming midterm or final exam. Don t delay! Go to right now to see the date of the next seminar. A seminar is generally held one or two weeks before the exam, but don t risk missing it just because you didn t check the date well in adance. You can also resere your place at the seminar online. You are not obligated to attend if you resere a place. You only pay for the seminar if and when you arrie. Grant s Weekly Tutoring Groups This is for the student who wants extra motiation and help keeping on top of things throughout the course. Again, go to for more details on when the groups are and how they work. Grant s Audio Lectures For less than the cost of 2 hours of one-on-one tutoring, you can listen to oer 40 hours of Grant teaching this book. Hear Grant work through examples, and offer that extra bit of explanation beyond the written word. Go to for more details. Grant Skene for Grant s Tutoring (text or call (204) ) DO NOT RECOPY Grant s Tutoring is a priate tutoring organization and is in no way affiliated with the Uniersity of Manitoba.

5 TABLE OF CONTENTS FOR VOLUME 1 (These Lessons are in Volume 1) Lesson 1: Systems of Linear Equations The Lecture... 1 The Lecture Problems Lesson 2: Row-Reduction and Linear Systems Lecture Problems The Lecture Homework and Practise Problems Solutions to Practise Problems Lesson 3: Matrix Math Important Matrix Facts and Definitions Lecture Problems The Lecture Homework and Practise Problems Solutions to Practise Problems Lesson 4: The Inerse of a Matrix and Applications Lecture Problems The Lecture Homework and Practise Problems Solutions to Practise Problems Lesson 5: Elementary Matrices Lecture Problems The Lecture Homework and Practise Problems Solutions to Practise Problems Lesson 6: Determinants and Their Properties Important Determinant Facts and Properties Lecture Problems The Lecture Homework and Practise Problems Solutions to Practise Problems Lesson 7: The Adjoint Matrix Lecture Problems The Lecture Homework and Practise Problems Solutions to Practise Problems Lesson 8: Cramer s Rule Lecture Problems The Lecture Homework and Practise Problems Solutions to Practise Problems THE MIDTERM EXAM NORMALLY COVERS LESSONS 1 TO 8. Grant Skene for Grant s Tutoring ( DO NOT RECOPY

6 TABLE OF CONTENTS FOR VOLUME 2 (These Lessons are in Volume 2) Lesson 9: Vectors Important Vector Formulas and Facts Lecture Problems The Lecture Homework and Practise Problems Solutions to Practise Problems Lesson 10: Lines and Planes Equations of Lines and Planes, etc Lecture Problems The Lecture Homework and Practise Problems Solutions to Practise Problems Lesson 11: Vector Spaces and Subspaces The Definition of Subspace and The 10 Axioms of Vector Space Lecture Problems The Lecture Homework and Practise Problems Solutions to Practise Problems Lesson 12: Linear Independence Key Definitions and Facts Lecture Problems The Lecture Homework and Practise Problems Solutions to Practise Problems Lesson 13: Basis and Dimension The Definition of Basis Lecture Problems The Lecture Homework and Practise Problems Solutions to Practise Problems Lesson 14: Marko Analysis Important Facts about Marko Analysis Lecture Problems The Lecture Homework and Practise Problems Solutions to Practise Problems Lesson 15: Linear Transformations Important Facts and Defintions about Linear Transformations Lecture Problems The Lecture Lesson 16: Eigenalues & Eigenectors Important Facts and Defintions about Eigenalues & Eigenectors Lecture Problems The Lecture THE MIDTERM EXAM NORMALLY COVERS LESSONS 1 TO 8. Grant Skene for Grant s Tutoring (phone (204) ) DO NOT RECOPY

7 (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 a b c d 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 Which is to say, the angle,, between the two ectors, u and, is an acute angle. } If u is negatie, then Which is to say, the angle,, between the two ectors, u and, is an obtuse angle. SAMPLE Grant Skene for Grant s Tutoring ( DO NOT RECOPY

8 274 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 Grant Skene for Grant s Tutoring (text or call (204) ) DO NOT RECOPY

9 (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 Grant Skene for Grant s Tutoring ( DO NOT RECOPY

10 276 LESSON 9: VECTORS (Linear Algebra) SAMPLE Grant Skene for Grant s Tutoring (text or call (204) ) DO NOT RECOPY