linear algebra...linear algebra and vector geometry (math 1300) matrices for management (math 1310)...

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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 volumes.

© Grant Skene for Grant’s Tutoring (text or call (204) 489-2884) DO NOT RECOPY


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HOW TO USE THIS BOOK

I have 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 solve all the Practise Problems I have given you, then you should have

nothing to fear about your Midterm or Final Exam.

I have presented the course in what I consider to be the most logical order. Although my

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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.

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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 Inverse of a Matrix and Applications

Lecture Problems ...................................................................................................................................................................................... 126

The Lecture ............................................................................................................................................................................................... 128



Lesson 5: Elementary Matrices


The Lecture ............................................................................................................................................................................................... 160



Lesson 6: Determinants and Their Properties

Important Determinant Facts and Properties. ........................................................................................................................................... 187


The Lecture ............................................................................................................................................................................................... 191



Lesson 7: The Adjoint Matrix


The Lecture ............................................................................................................................................................................................... 237



Lesson 8: Cramer’s Rule


The Lecture ............................................................................................................................................................................................... 265



THE MIDTERM EXAM NORMALLY COVERS LESSONS 1 TO 8.

© Grant Skene for Grant’s Tutoring (phone (204) 489-2884) 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



Lesson 10: Lines and Planes

Equations of Lines and Planes, etc. ............................................................................................................................................................ 325


The Lecture ................................................................................................................................................................................................ 327



Lesson 11: Vector Spaces and Subspaces

The Definition of Subspace and The 10 Axioms of Vector Space. ............................................................................................................... 358


The Lecture ................................................................................................................................................................................................ 361



Lesson 12: Linear Independence

Key Definitions and Facts. .......................................................................................................................................................................... 398


The Lecture ................................................................................................................................................................................................ 400



Lesson 13: Basis and Dimension

The Definition of Basis. .............................................................................................................................................................................. 433


The Lecture ................................................................................................................................................................................................ 436



Lesson 14: Markov Analysis Important Facts about Markov Analysis ..................................................................................................................................................... 467


The Lecture ................................................................................................................................................................................................ 470



Lesson 15: Linear Transformations Important Facts and Defintions about Linear Transformations ................................................................................................................ 15-1

Lecture Problems ................................................................................................................................................................................... 15-10

The Lecture ............................................................................................................................................................................................ 15-13

Lesson 16: Eigenvalues & Eigenvectors Important Facts and Defintions about Eigenvalues & Eigenvectors ......................................................................................................... 16-1

Lecture Problems ..................................................................................................................................................................................... 16-2

The Lecture .............................................................................................................................................................................................. 16-3

THE MIDTERM EXAM NORMALLY COVERS LESSONS 1 TO 8.

(Linear Algebra) LESSON 9: VECTORS 273

© 1997-2011 Grant Skene for Grant’s Tutoring (www.grantstutoring.com) DO NOT RECOPY

Lesson 9: Vectors

Important Vector Formulas and Facts:

} If v is the vector (a, b, c), which could also be denoted ai + bj + ck, then the length

of v (also called the norm of v) is denoted v , where:

2 2 2

a b c v

} Similarly, if v = (a, b, c, d), then 2 2 2 2

a b c d v . The pattern holds for

vectors of any size, Rn

.

} A unit vector is a vector whose length is exactly 1 unit (i.e., the norm of

the vector is 1). Three standard unit vectors are i, the unit vector in the positive x

direction, j, the unit vector in the positive y direction, and k, the unit vector in the

positive z direction.

} The unit vector in the direction of any vector v can be denoted v̂ , where:

1ˆ v v

v

} Given two vectors, u and v, where the dot product 0u v , then u is orthogonal

to v (u and v make a right angle).

} The cosine of the angle, , between two vectors, u and v, is given by:

cos u v

u v

} If u v is positive, then 0 90 . Which is to say, the angle, , between the

two vectors, u and v, is an acute angle.

} If u v is negative, then 90 180 . Which is to say, the angle, , between the

two vectors, u and v, is an obtuse angle.

SAM

PLE

274 LESSON 9: VECTORS (Linear Algebra)

© 1997-2011 Grant Skene for Grant’s Tutoring (text or call (204) 489-2884) DO NOT RECOPY

} Given two vectors, u and v, the orthogonal projection of u onto v is denoted projv

u ,

where:

2proj

v

u vu v

v

} Given two vectors, u and v, the cross product, u v , produces a third vector, w,

which is orthogonal to both u and v. In fact, w is normal to the entire plane

containing u and v (i.e., w makes a right angle with the u, v plane).

} u v v u but u v v u ; however, u v v u .

} The sine of the angle, , between two vectors, u and v, is given by:

sin

u v

u v

} The area of a parallelogram with adjacent sides defined by the vectors u and v is

given by:

Area of a parallelogram = u v

} The area of a triangle with two sides defined by the vectors u and v is given by:

Area of a triangle = 1

2

u v

} The volume of a parallelepiped with adjacent edges defined by the vectors u, v and

w is given by:

Volume of a parallelepiped = u v w

(Actually, you can pick any pair you want to compute the cross product with,

then compute the dot product with the remaining vector.

The important thing is do the cross product first, then the dot product.)

Volume of the parallelepiped = u w v also, or

Volume of the parallelepiped = v w u also.

SAM

PLE

(Linear Algebra) LESSON 9: VECTORS 275

© 1997-2011 Grant Skene for Grant’s Tutoring (www.grantstutoring.com) DO NOT RECOPY

Lecture Problems:

(Each of the questions below will be discussed and solved in the lecture that follows.)

1. Let u=(3, –1, 4) and v=(–2, 1, –3). Find the following:

(a) u (b) u v (c) u × v (d) 2 3u v

(e) The cosine of the angle between u and v.

(f) A unit vector in the direction of u.

(g) The projection of u onto v.

(h) A vector, w, which is orthogonal to u and v.

(i) A vector, m, which is orthogonal to u but not orthogonal to v.

(j) The area of the triangle with adjacent sides u and v.

2. Let v=(2, 7, 1). Find a vector that is 10 units long, but pointing in the exact opposite

direction to v.

3. (a) Let v1 = (−1, 2, −2, −4), v

2 = (3, 1, 1, 3), and v

3 = (p, 3, q, 4).

Find p and q such that v3 is orthogonal to both v

1 and v

2.

(b) Let u = (1, 2, 3, 4) and v = (2, 5, 6, 7).

Find a non-zero R4

vector w that is orthogonal to both u and v.

4. Given 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 vertices of

parallelogram ABCD. Use vectors to find the area of that parallelogram.

6. ABCD is a parallelogram. Let u = AD and v = AB .

(a) Write the vector AC as a linear combination of

the vectors u and v.

(b) Write the vector BD as a linear combination of

the vectors u and v.

(c) Use vector 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 vectors to find the distance between the point P = (4, 5) and the line 2x + 3y = 6.

B A

C D

SAM

PLE

276 LESSON 9: VECTORS (Linear Algebra)

© 1997-2011 Grant Skene for Grant’s Tutoring (text or call (204) 489-2884) DO NOT RECOPY

SAM

PLE

linear algebra...linear algebra and vector geometry (math 1300) matrices for management (math 1310)...

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