# Introduction to Linear Algebra, 2nd edition

CRC Press, Apr 1, 1991 - Mathematics - 288 pages
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
In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n 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 n×n matrices over any commutative ring R. In particular, this function exists when R is the field of real or complex numbers.
Contents
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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
12 References
 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.
 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 2×2 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.
 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 3×3 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.
 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 Copyright