MATH1051
                                                                                                    CALCULUS
                                                                                                                       AND
                                                                           LINEAR ALGEBRA I
                                                                                                 Semester 1, 2008 
                                                                        Lecture Workbook Solutions
                                                How to use this workbook
                                                This book should be taken to lectures, tutorials and lab sessions.  In the lectures, 
                                                you will be expected to fill in the blanks in any incomplete definitions, theorems 
                                                etc, and any examples which appear. The lecturer will make it clear when they 
                                                are covering something that you need to complete.
                                                The completed workbook will act as a study guide designed to assist you in 
                                                working through assignments and preparing for the mid-semester and final ex-
                                                ams. For this reason, it is very important to attend lectures.
                                                The text for the calculus part of the course is “Calculus” by James Stewart, 
                                                6th edition, 2008. We often refer to the text in this workbook, and many of the 
                                                definitions, theorems, and examples come from the text.  (References are also 
                                                provided for the 5th edition).  It is strongly advised that you purchase a copy.
                                                Although there is no set text for the linear algebra part of the course,  there 
                                                are several books in the library which cover the material well (see the course 
                                                website).
                                                 c 
                                                
   Mathematics, School of Physical Sciences, The University of Queensland, Brisbane QLD 4072, Australia 
                                                Edited by Phil Isaac, Birgit Loch, Joseph Grotowski, Mary Waterhouse, Victor Scharaschkin, Barbara Maen-
                                                haut.  July 2007.
                                                                                                                                   1
                         Contents
                         1 Complex Numbers                                                                   9
                            1.1   Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     9
                            1.2   Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . .          9
                            1.3   Polar form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    10
                            1.4   Euler’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . .     12
                         2 Vectors                                                                         13
                            2.1   Revision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    13
                            2.2   Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . .       20
                         3 Matrices and Linear Transformations                                             24
                            3.1   2×2matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . .       24
                            3.2   Transformations of the plane . . . . . . . . . . . . . . . . . . . .      24
                            3.3   Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    25
                            3.4   Linear transformations . . . . . . . . . . . . . . . . . . . . . . .      26
                            3.5   Visualizing linear transformations . . . . . . . . . . . . . . . . .      27
                            3.6   Scalings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    28
                            3.7   Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     30
                            3.8   Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    31
                            3.9   Composing Linear Transformations; Matrix Multiplication . . .             34
                            3.10 Vectors in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . .      35
                            3.11 Properties of Dot Product . . . . . . . . . . . . . . . . . . . . .        37
                            3.12 Definition: Matrix       . . . . . . . . . . . . . . . . . . . . . . . . .  37
                            3.13 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     38
                            3.14 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     38
                            3.15 Scalar Multiplication . . . . . . . . . . . . . . . . . . . . . . . .      38
                            3.16 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . .      39
                            3.17 Transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . .      41
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                          4 Vector Spaces                                                                      43
                             4.1   Linear combinations      . . . . . . . . . . . . . . . . . . . . . . . .    43
                             4.2   Linear Independence . . . . . . . . . . . . . . . . . . . . . . . .         44
                             4.3   How to test for linear independence . . . . . . . . . . . . . . . .         44
                             4.4   Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .       47
                             4.5   The span of vectors . . . . . . . . . . . . . . . . . . . . . . . . .       50
                             4.6   Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     52
                             4.7   Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       54
                             4.8   Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        55
                             4.9   Further Properties of Bases       . . . . . . . . . . . . . . . . . . . .   56
                             4.10 Orthogonal and Orthonormal . . . . . . . . . . . . . . . . . . .             57
                             4.11 Gram-Schmidt Algorithm . . . . . . . . . . . . . . . . . . . . .             59
                          5 Inverses                                                                           62
                             5.1   The Identity Matrix      . . . . . . . . . . . . . . . . . . . . . . . .    62
                             5.2   Definition: Inverse     . . . . . . . . . . . . . . . . . . . . . . . . .    63
                             5.3   Invertible Matrices and Linear Independence . . . . . . . . . . .           65
                          6 Gaussian elimination                                                               71
                             6.1   Simultaneous equations . . . . . . . . . . . . . . . . . . . . . . .        71
                             6.2   Gaussian elimination . . . . . . . . . . . . . . . . . . . . . . . .        72
                             6.3   The general solution     . . . . . . . . . . . . . . . . . . . . . . . .    82
                             6.4   Gauss-Jordan elimination       . . . . . . . . . . . . . . . . . . . . .    85
                          7 Determinants                                                                       90
                             7.1   Definition: Determinant . . . . . . . . . . . . . . . . . . . . . .          90
                             7.2   Properties of Determinants . . . . . . . . . . . . . . . . . . . . .        92
                             7.3   Connection with inverses and systems of linear equations . . . .            97
                          8 Vector Products in 3-Space                                                         99
                             8.1   Definition: Cross product       . . . . . . . . . . . . . . . . . . . . .    99
                             8.2   Application: area of a triangle . . . . . . . . . . . . . . . . . . .      102
                             8.3   Scalar Triple Product . . . . . . . . . . . . . . . . . . . . . . . .      104
                             8.4   Geometrical Interpretation . . . . . . . . . . . . . . . . . . . . .       104
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                          9 Eigenvalues and eigenvectors                                                     106
                             9.1   Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      106
                             9.2   Eigenspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     107
                             9.3   How to find eigenvalues . . . . . . . . . . . . . . . . . . . . . . .       107
                          10 Numbers                                                                         114
                             10.1 Number systems        . . . . . . . . . . . . . . . . . . . . . . . . . .   114
                             10.2 Real number line and ordering on R . . . . . . . . . . . . . . . .          114
                             10.3 Definition: Intervals . . . . . . . . . . . . . . . . . . . . . . . . .      115
                             10.4 Absolute value      . . . . . . . . . . . . . . . . . . . . . . . . . . .   116
                          11 Functions                                                                       119
                             11.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      119
                             11.2 Definition: Function, domain, range . . . . . . . . . . . . . . . .          119
                             11.3 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      120
                             11.4 Convention (domain) . . . . . . . . . . . . . . . . . . . . . . . .         121
                             11.5 Caution (non-functions) . . . . . . . . . . . . . . . . . . . . . .         121
                             11.6 Vertical line test . . . . . . . . . . . . . . . . . . . . . . . . . . .    122
                             11.7 Exponential functions . . . . . . . . . . . . . . . . . . . . . . . .       123
                             11.8 Trigonometric functions (sin, cos, tan)        . . . . . . . . . . . . . .  126
                             11.9 Composition of functions . . . . . . . . . . . . . . . . . . . . . .        131
                             11.10One-to-one (1-1) functions - Stewart, 6ed.           pp. 385-388; 5ed.
                                   pp. 413-417 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      131
                             11.11Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . .       133
                             11.12Logarithms - Stewart, 6ed. p. 405; 5ed. p. 434 . . . . . . . . . .          135
                             11.13Natural logarithm . . . . . . . . . . . . . . . . . . . . . . . . . .       135
                             11.14Inverse trigonometric functions        . . . . . . . . . . . . . . . . . .  138
                          12 Limits                                                                          146
                             12.1 Definition: Limit - Stewart, 6ed. p. 66; 5ed. p. 71 . . . . . . . .          146
                             12.2 One-sided limits . . . . . . . . . . . . . . . . . . . . . . . . . . .      150
                             12.3 Theorem: Squeeze principle . . . . . . . . . . . . . . . . . . . .          152
                             12.4 Limits as x approaches infinity . . . . . . . . . . . . . . . . . . .        153
                             12.5 Some important limits . . . . . . . . . . . . . . . . . . . . . . .         157
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