A First Course in Linear Algebra

✍ Scribed by Robert A Beezer

• Tongue: English
• Leaves: 848
• Category: Library

✦ Synopsis

A First Course in Linear Algebra is an introduction to the basic concepts of linear algebra, along with an introduction to the techniques of formal mathematics. It begins with systems of equations and matrix algebra before moving into the theory of abstract vector spaces, eigenvalues, linear transformations and matrix representations. It has numerous worked examples and exercises, along with precise statements of definitions and complete proofs of every theorem, making it ideal for independent study.

✦ Table of Contents

Preface
Contents
Definitions
Theorems
Notation
Examples
Proof Techniques
Computation Notes
Contributors
GNU Free Documentation License
1. APPLICABILITY AND DEFINITIONS
2. VERBATIM COPYING
3. COPYING IN QUANTITY
4. MODIFICATIONS
5. COMBINING DOCUMENTS
6. COLLECTIONS OF DOCUMENTS
7. AGGREGATION WITH INDEPENDENT WORKS
8. TRANSLATION
9. TERMINATION
10. FUTURE REVISIONS OF THIS LICENSE
ADDENDUM: How to use this License for your documents
Part C Core
Chapter SLE Systems of Linear Equations
WILA What is Linear Algebra?
LA Linear'' +Algebra''
A An application: packaging trail mix
READ Reading Questions
EXC Exercises
SOL Solutions
SSLE Solving Systems of Linear Equations
PSS Possibilities for solution sets
ESEO Equivalent systems and equation operations
READ Reading Questions
EXC Exercises
SOL Solutions
RREF Reduced Row-Echelon Form
READ Reading Questions
EXC Exercises
SOL Solutions
TSS Types of Solution Sets
READ Reading Questions
EXC Exercises
SOL Solutions
HSE Homogeneous Systems of Equations
SHS Solutions of Homogeneous Systems
MVNSE Matrix and Vector Notation for Systems of Equations
NSM Null Space of a Matrix
READ Reading Questions
EXC Exercises
SOL Solutions
NSM NonSingular Matrices
NSM NonSingular Matrices
READ Reading Questions
EXC Exercises
SOL Solutions
Chapter V Vectors
VO Vector Operations
VEASM Vector equality, addition, scalar multiplication
VSP Vector Space Properties
READ Reading Questions
EXC Exercises
SOL Solutions
LC Linear Combinations
LC Linear Combinations
VFSS Vector Form of Solution Sets
PSHS Particular Solutions, Homogeneous Solutions
URREF Uniqueness of Reduced Row-Echelon Form
READ Reading Questions
EXC Exercises
SOL Solutions
SS Spanning Sets
SSV Span of a Set of Vectors
SSNS Spanning Sets of Null Spaces
READ Reading Questions
EXC Exercises
SOL Solutions
LI Linear Independence
LISV Linearly Independent Sets of Vectors
LINSM Linear Independence and NonSingular Matrices
NSSLI Null Spaces, Spans, Linear Independence
READ Reading Questions
EXC Exercises
SOL Solutions
LDS Linear Dependence and Spans
LDSS Linearly Dependent Sets and Spans
COV Casting Out Vectors
READ Reading Questions
EXC Exercises
SOL Solutions
O Orthogonality
CAV Complex arithmetic and vectors
IP Inner products
N Norm
OV Orthogonal Vectors
GSP Gram-Schmidt Procedure
READ Reading Questions
EXC Exercises
Chapter M Matrices
MO Matrix Operations
MEASM Matrix equality, addition, scalar multiplication
VSP Vector Space Properties
TSM Transposes and Symmetric Matrices
MCC Matrices and Complex Conjugation
READ Reading Questions
EXC Exercises
SOL Solutions
MM Matrix Multiplication
MVP Matrix-Vector Product
MM Matrix Multiplication
MMEE Matrix Multiplication, Entry-by-Entry
PMM Properties of Matrix Multiplication
READ Reading Questions
EXC Exercises
SOL Solutions
MISLE Matrix Inverses and Systems of Linear Equations
IM Inverse of a Matrix
CIM Computing the Inverse of a Matrix
PMI Properties of Matrix Inverses
READ Reading Questions
EXC Exercises
SOL Solutions
MINSM Matrix Inverses and NonSingular Matrices
NSMI NonSingular Matrices are Invertible
OM Orthogonal Matrices
READ Reading Questions
EXC Exercises
SOL Solutions
CRS Column and Row Spaces
CSSE Column spaces and systems of equations
CSSOC Column space spanned by original columns
CSNSM Column Space of a Nonsingular Matrix
RSM Row Space of a Matrix
READ Reading Questions
EXC Exercises
SOL Solutions
FS Four Subsets
LNS Left Null Space
CRS Computing Column Spaces
EEF Extended echelon form
FS Four Subsets
READ Reading Questions
EXC Exercises
SOL Solutions
Chapter VS Vector Spaces
VS Vector Spaces
VS Vector Spaces
EVS Examples of Vector Spaces
VSP Vector Space Properties
RD Recycling Definitions
READ Reading Questions
EXC Exercises
S Subspaces
TS Testing Subspaces
TSS The Span of a Set
SC Subspace Constructions
READ Reading Questions
EXC Exercises
SOL Solutions
LISS Linear Independence and Spanning Sets
LI Linear independence
SS Spanning Sets
VR Vector Representation
READ Reading Questions
EXC Exercises
SOL Solutions
B Bases
B Bases
BSCV Bases for Spans of Column Vectors
BNSM Bases and NonSingular Matrices
OBC Orthonormal Bases and Coordinates
READ Reading Questions
EXC Exercises
SOL Solutions
D Dimension
D Dimension
DVS Dimension of Vector Spaces
RNM Rank and Nullity of a Matrix
RNNSM Rank and Nullity of a NonSingular Matrix
READ Reading Questions
EXC Exercises
SOL Solutions
PD Properties of Dimension
GT Goldilocks' Theorem
RT Ranks and Transposes
DFS Dimension of Four Subspaces
READ Reading Questions
EXC Exercises
SOL Solutions
Chapter D Determinants
DM Determinants of Matrices
EM Elementary Matrices
DD Definition of the Determinant
CD Computing Determinants
READ Reading Questions
EXC Exercises
SOL Solutions
PDM Properties of Determinants of Matrices
DRO Determinants and Row Operations
DROEM Determinants, Row Operations, Elementary Matrices
DNMMM Determinants, Nonsingular Matrices, Matrix Multiplication
READ Reading Questions
EXC Exercises
SOL Solutions
Chapter E Eigenvalues
EE Eigenvalues and Eigenvectors
EEM Eigenvalues and Eigenvectors of a Matrix
PM Polynomials and Matrices
EEE Existence of Eigenvalues and Eigenvectors
CEE Computing Eigenvalues and Eigenvectors
ECEE Examples of Computing Eigenvalues and Eigenvectors
READ Reading Questions
EXC Exercises
SOL Solutions
PEE Properties of Eigenvalues and Eigenvectors
ME Multiplicities of Eigenvalues
EHM Eigenvalues of Hermitian Matrices
READ Reading Questions
EXC Exercises
SOL Solutions
SD Similarity and Diagonalization
SM Similar Matrices
PSM Properties of Similar Matrices
D Diagonalization
OD Orthonormal Diagonalization
READ Reading Questions
EXC Exercises
SOL Solutions
Chapter LT Linear Transformations
LT Linear Transformations
LT Linear Transformations
MLT Matrices and Linear Transformations
LTLC Linear Transformations and Linear Combinations
PI Pre-Images
NLTFO New Linear Transformations From Old
READ Reading Questions
EXC Exercises
SOL Solutions
ILT Injective Linear Transformations
EILT Examples of Injective Linear Transformations
KLT Kernel of a Linear Transformation
ILTLI Injective Linear Transformations and Linear Independence
ILTD Injective Linear Transformations and Dimension
CILT Composition of Injective Linear Transformations
READ Reading Questions
EXC Exercises
SOL Solutions
SLT Surjective Linear Transformations
ESLT Examples of Surjective Linear Transformations
RLT Range of a Linear Transformation
SSSLT Spanning Sets and Surjective Linear Transformations
SLTD Surjective Linear Transformations and Dimension
CSLT Composition of Surjective Linear Transformations
READ Reading Questions
EXC Exercises
SOL Solutions
IVLT Invertible Linear Transformations
IVLT Invertible Linear Transformations
IV Invertibility
SI Structure and Isomorphism
RNLT Rank and Nullity of a Linear Transformation
SLELT Systems of Linear Equations and Linear Transformations
READ Reading Questions
EXC Exercises
SOL Solutions
Chapter R Representations
VR Vector Representations
CVS Characterization of Vector Spaces
CP Coordinatization Principle
READ Reading Questions
EXC Exercises
SOL Solutions
MR Matrix Representations
NRFO New Representations from Old
PMR Properties of Matrix Representations
IVLT Invertible Linear Transformations
READ Reading Questions
EXC Exercises
SOL Solutions
CB Change of Basis
EELT Eigenvalues and Eigenvectors of Linear Transformations
CBM Change-of-Basis Matrix
MRS Matrix Representations and Similarity
CELT Computing Eigenvectors of Linear Transformations
READ Reading Questions
EXC Exercises
SOL Solutions
Chapter A Archetypes
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Part T Topics
Chapter P Preliminaries
CNO Complex Number Operations
CNA Arithmetic with complex numbers
CCN Conjugates of Complex Numbers
MCN Modulus of a Complex Number
Part A Applications