Introduction to Linear Algebra, 2nd edition

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CRC Press, Apr 1, 1991 - Mathematics - 288 pages
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This popular textbook was thoughtfully and specifically tailored to introducing undergraduate students to linear algebra. The second edition has been carefully revised to improve upon its already successful format and approach. In particular, the author added a chapter on quadratic forms, making this one of the most comprehensive introductory texts on linear algebra.
 

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Determinant
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In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nn square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. Determinants are important both in calculus, where they enter the substitution rule for several variables, and in multilinear algebra.
For a fixed positive integer n, there is a unique determinant function for the nn matrices over any commutative ring R. In particular, this function exists when R is the field of real or complex numbers.
Contents
[hide]
1 Vertical bar notation
2 Determinants of 2-by-2 matrices
3 Determinants of 3-by-3 matrices
4 Applications
5 General definition and computation
6 Example
7 Properties
7.1 Useful identities
7.2 Block matrices
7.3 Relationship to trace
7.4 Derivative
8 Abstract formulation
9 Algorithmic implementation
10 History
11 See also
12 References
13 External links
[edit] Vertical bar notation
The determinant of a matrix A is also sometimes denoted by |A|. This notation can be ambiguous since it is also used for certain matrix norms and for the absolute value. However, often the matrix norm will be denoted with double vertical bars (e.g., ||A||) and may carry a subscript as well. Thus, the vertical bar notation for determinant is frequently used (e.g., Cramer's rule and minors).
For example, for matrix
the determinant det(A) might be indicated by | A | or more explicitly as
That is, the square braces around the matrices are replaced with elongated vertical bars.
[edit] Determinants of 2-by-2 matrices
The area of the parallelogram is the determinant of the matrix formed by the vectors representing the parallelogram's sides.The 22 matrix,
has determinant
The interpretation when the matrix has real number entries is that this gives the oriented area of the parallelogram with vertices at (0,0), (a,b), (a + c, b + d), and (c,d). The oriented area is the same as the usual area, except that it is negative when the vertices are listed in clockwise order.
The assumption here is that a linear transformation is applied to row vectors as the vector-matrix product xTAT, where x is a column vector. The parallelogram in the figure is obtained by multiplying matrix A (which stores the co-ordinates of our parallelogram) with each of the row vectors and in turn. These row vectors define the vertices of the unit square. With the more common matrix-vector product Ax, the parallelogram has vertices at and (note that Ax = (xTAT)T).
A formula for larger matrices will be given below.
[edit] Determinants of 3-by-3 matrices
The volume of this Parallelepiped is the absolute value of the determinant of the matrix formed by the rows r1, r2, and r3.The 33 matrix:
Using the cofactor expansion on the first row of the matrix we get:
The determinant of a 3x3 matrix can be calculated by its diagonals.which can be remembered as the sum of the products of three diagonal north-west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements when the copies of the first two columns of the matrix are written beside it as below:
Note that this mnemonic does not carry over into higher dimensions.
[edit] Applications
Determinants are used to characterize invertible matrices (i.e., exactly those matrices with non-zero determinants), and to explicitly describe the solution to a system of linear equations with Cramer's rule. They can be used to find the eigenvalues of the matrix A through the characteristic polynomial
where I is the identity matrix of the same dimension as A.
One often thinks of the determinant as assigning a number to every sequence of n vectors in , by using the square matrix whose columns
 

Contents

Chapter One A SYSTEM OF VECTORS 1 Introduction
1
Description of the system 3
2
Directed line segments and position vectors
3
Addition and subtraction of vectors
5
Multiplication of a vector by a scalar
8
Section formula and collinear points
10
Centroids of a triangle and a tetrahedron
12
Coordinates and components
14
Direct sums of subspaces
126
Exercises on chapter 5
130
Chapter Six LINEAR MAPPINGS 44 Introduction
133
Some examples of linear mappings
134
Some elementary facts about linear mappings
136
New linear mappings from old
138
Image space and kernel of a linear mapping
140
Rank and nullity
144

Scalar products
18
Postscript
22
Exercises on chapter 1
24
Chapter Two MATRICES 11 Introduction
26
Basic nomenclature for matrices
27
Addition and subtraction of matrices
29
Multiplication of a matrix by a scalar
30
Multiplication of matrices
31
Properties and nonproperties of matrix multiplication
33
Some special matrices and types of matrices
37
Transpose of a matrix
39
First considerations of matrix inverses
41
Properties of nonsingular matrices
43
Partitioned matrices
45
Exercises on chapter 2
49
Chapter Three ELEMENTARY ROW OPERATIONS 22 Introduction
52
Some generalities concerning elementary row operations
53
Echelon matrices and reduced echelon matrices
55
Elementary matrices
61
Major new insights on matrix inverses
63
Generalities about systems of linear equations
67
Elementary row operations and systems of linear equations
70
Exercises on chapter 3
77
Chapter Four AN INTRODUCTION TO DETERMINANTS 29 Preface to the chapter
79
Minors cofactors and larger determinants
80
Basic properties of determinants
83
The multiplicative property of determinants
87
Another method for inverting a nonsingular matrix
90
Exercises on chapter 4
92
Chapter Five VECTOR SPACES 34 Introduction
94
The definition of a vector space and examples
95
Elementary consequences of the vector space axioms
97
Subspaces
99
Spanning sequences
102
Linear dependence and independence
106
Bases and dimension
112
Further theorems about bases and dimension
118
Sums of subspaces
122
Rowand columnrank of a matrix
147
Systems of linear equations revisited
150
Rank inequalities
154
Vector spaces of linear mappings
158
Exercises on chapter 6
161
Chapter Seven MATRICES FROM LINEAR MAPPINGS 54 Introduction
164
The main definition and its immediate consequences
165
Matrices of sums etc of linear mappings
170
Change of basis
172
Matrix of a linear mapping w r t different bases
174
The simplest matrix representing a linear mapping
176
Vector space isomorphisms
178
Exercises on chapter 7
181
Chapter Eight EIGENVALUES EIGENVECTORS AND DIAGONALIZATION 61 Introduction
184
Eigenvalues and eigenvectors
187
Eigenvalues in the case F C
193
Diagonalization of linear transformations
196
Diagonalization of square matrices
199
The hermitian conjugate of a complex matrix
201
Eigenvalues of special types of matrices
202
Exercises on chapter 8
204
Chapter Nine EUCLIDEAN SPACES 69 Introduction
207
Some elementary results about euclidean spaces
209
Orthonormal sequences and bases
211
Lengthpreserving transformations of a euclidean space
215
Orthogonal diagonalization of a real symmetric matrix
219
Exercises on chapter 9
224
Chapter Ten QUADRATIC FORMS 74 Introduction
226
Change of basis and change of variable
230
Diagonalization of a quadratic form
233
Invariants of a quadratic form
239
Orthogonal diagonalization of a real quadratic form
242
Positivedefinite real quadratic forms
245
The leading minors theorem
249
vii
254
Appendix MAPPINGS
257
INDEX
273
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