Linear Algebra in 25 Lectures - Denton, Waldron - Creative Commons (2011).pdf

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LinearAlgebrainTwentyFiveLectures
Tom Denton and Andrew Waldron
March 27, 2011
Edited by Rohit Thomas
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Contents
1 What is Linear Algebra?
8
2 Gaussian Elimination 14
2.1 Notation for Linear Systems . . . . . . . . . . . . . . . . . . . 14
2.2 Reduced Row Echelon Form . . . . . . . . . . . . . . . . . . . 16
3 Elementary Row Operations
21
4 Solution Sets for Systems of Linear Equations 27
4.1 Non-Leading Variables . . . . . . . . . . . . . . . . . . . . . . 27
5 Vectors in Space, n-Vectors 36
5.1 Directions and Magnitudes . . . . . . . . . . . . . . . . . . . . 38
6 Vector Spaces
44
7 Linear Transformations
49
8 Matrices
54
9 Properties of Matrices 62
9.1 Block Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 62
9.2 The Algebra of Square Matrices . . . . . . . . . . . . . . . . 63
10 Inverse Matrix 68
10.1 Three Properties of the Inverse . . . . . . . . . . . . . . . . . 68
10.2 Finding Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . 69
10.3 Linear Systems and Inverses . . . . . . . . . . . . . . . . . . . 70
10.4 Homogeneous Systems . . . . . . . . . . . . . . . . . . . . . . 71
10.5 Bit Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
11 LU Decomposition 74
11.1 Using LU Decomposition to Solve Linear Systems . . . . . . . 75
11.2 Finding an LU Decomposition. . . . . . . . . . . . . . . . . . 76
11.3 Block LU Decomposition . . . . . . . . . . . . . . . . . . . . . 79
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12 Elementary Matrices and Determinants 81
12.1 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
12.2 Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . . 84
13 Elementary Matrices and Determinants II
88
14 Properties of the Determinant 94
14.1 Determinant of the Inverse . . . . . . . . . . . . . . . . . . . . 97
14.2 Adjoint of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . 97
14.3 Application: Volume of a Parallelepiped . . . . . . . . . . . . 99
15 Subspaces and Spanning Sets 102
15.1 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
15.2 Building Subspaces . . . . . . . . . . . . . . . . . . . . . . . . 103
16 Linear Independence
108
17 Basis and Dimension
115
17.1 Bases inR
n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
18 Eigenvalues and Eigenvectors 122
18.1 Invariant Directions . . . . . . . . . . . . . . . . . . . . . . . . 122
19 Eigenvalues and Eigenvectors II 128
19.1 Eigenspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
20 Diagonalization 134
20.1 Matrix of a Linear Transformation . . . . . . . . . . . . . . . 134
20.2 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . 137
20.3 Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 138
21 Orthonormal Bases 143
21.1 Relating Orthonormal Bases . . . . . . . . . . . . . . . . . . . 146
22 Gram-Schmidt and Orthogonal Complements 151
22.1 Orthogonal Complements . . . . . . . . . . . . . . . . . . . . 155
23 Diagonalizing Symmetric Matrices
160
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24 Kernel, Range, Nullity, Rank 166
24.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
25 Least Squares
174
A Sample Midterm I Problems and Solutions
178
B Sample Midterm II Problems and Solutions
188
C Sample Final Problems and Solutions
198
Index
223
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Preface
These linear algebra lecture notes are designed to be presented as twenty ve,
fty minute lectures suitable for sophomores likely to use the material for
applications but still requiring a solid foundation in this fundamental branch
of mathematics. The main idea of the course is to emphasize the concepts
of vector spaces and linear transformations as mathematical structures that
can be used to model the world around us. Once \persuaded" of this truth,
students learn explicit skills such as Gaussian elimination and diagonalization
in order that vectors and linear transformations become calculational tools,
rather than abstract mathematics.
In practical terms, the course aims to produce students who can perform
computations with large linear systems while at the same time understand
the concepts behind these techniques. Often-times when a problem can be re-
duced to one of linear algebra it is \solved". These notes do not devote much
space to applications (there are already a plethora of textbooks with titles
involving some permutation of the words \linear", \algebra" and \applica-
tions"). Instead, they attempt to explain the fundamental concepts carefully
enough that students will realize for their own selves when the particular
application they encounter in future studies is ripe for a solution via linear
algebra.
The notes are designed to be used in conjunction with a set of online
homework exercises which help the students read the lecture notes and learn
basic linear algebra skills. Interspersed among the lecture notes are links
to simple online problems that test whether students are actively reading
the notes. In addition there are two sets of sample midterm problems with
solutions as well as a sample nal exam. There are also a set of ten online
assignments which are collected weekly. The rst assignment is designed to
ensure familiarity with some basic mathematic notions (sets, functions, logi-
cal quantiers and basic methods of proof). The remaining nine assignments
are devoted to the usual matrix and vector gymnastics expected from any
sophomore linear algebra class. These exercises are all available at
Webwork is an open source, online homework system which originated at
the University of Rochester. It can eciently check whether a student has
answered an explicit, typically computation-based, problem correctly. The
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