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A First Course in Linear Algebra
Robert A. Beezer
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Front Matter
Colophon
Author Biography
Dedication
Acknowledgements
Contributors
Preface
SLE
Systems of Linear Equations
WILA
What is Linear Algebra?
LA
Linear and Algebra
AA
An Application
WILA
Reading Questions
WILA
Exercises
SSLE
Solving Systems of Linear Equations
SLE
Systems of Linear Equations
PSS
Possibilities for Solution Sets
ESEO
Equivalent Systems and Equation Operations
SSLE
Reading Questions
SSLE
Exercises
RREF
Reduced Row-Echelon Form
MVNSE
Matrix and Vector Notation for Systems of Equations
RO
Row Operations
RREF
Reduced Row-Echelon Form
RREF
Reading Questions
RREF
Exercises
TSS
Types of Solution Sets
CS
Consistent Systems
FV
Free Variables
TSS
Reading Questions
TSS
Exercises
HSE
Homogeneous Systems of Equations
SHS
Solutions of Homogeneous Systems
NSM
Null Space of a Matrix
HSE
Reading Questions
HSE
Exercises
NM
Nonsingular Matrices
NM
Nonsingular Matrices
NSNM
Null Space of a Nonsingular Matrix
NM
Reading Questions
NM
Exercises
V
Vectors
VO
Vector Operations
CV
Column Vectors
VEASM
Vector Equality, Addition, Scalar Multiplication
VSP
Vector Space Properties
VO
Reading Questions
VO
Exercises
LC
Linear Combinations
LC
Linear Combinations
VFSS
Vector Form of Solution Sets
PSHS
Particular Solutions, Homogeneous Solutions
LC
Reading Questions
LC
Exercises
SS
Spanning Sets
SSV
Span of a Set of Vectors
SSNS
Spanning Sets of Null Spaces
SS
Reading Questions
SS
Exercises
LI
Linear Independence
LISV
Linearly Independent Sets of Vectors
LINM
Linear Independence and Nonsingular Matrices
NSSLI
Null Spaces, Spans, Linear Independence
LI
Reading Questions
LI
Exercises
LDS
Linear Dependence and Spans
LDSS
Linearly Dependent Sets and Spans
COV
Casting Out Vectors
LDS
Reading Questions
LDS
Exercises
O
Orthogonality
CAV
Complex Arithmetic and Vectors
IP
Inner products
N
Norm
OV
Orthogonal Vectors
GSP
Gram-Schmidt Procedure
O
Reading Questions
O
Exercises
M
Matrices
MO
Matrix Operations
MEASM
Matrix Equality, Addition, Scalar Multiplication
VSP
Vector Space Properties
TSM
Transposes and Symmetric Matrices
TM
Triangular Matrices
MCC
Matrices and Complex Conjugation
AM
Adjoint of a Matrix
MO
Reading Questions
MO
Exercises
MM
Matrix Multiplication
MVP
Matrix-Vector Product
MM
Matrix Multiplication
MMEE
Matrix Multiplication, Entry-by-Entry
PMM
Properties of Matrix Multiplication
HM
Hermitian Matrices
MM
Reading Questions
MM
Exercises
MISLE
Matrix Inverses and Systems of Linear Equations
SI
Solutions and Inverses
IM
Inverse of a Matrix
CIM
Computing the Inverse of a Matrix
PMI
Properties of Matrix Inverses
MISLE
Reading Questions
MISLE
Exercises
MINM
Matrix Inverses and Nonsingular Matrices
NMI
Nonsingular Matrices are Invertible
UM
Unitary Matrices
MINM
Reading Questions
MINM
Exercises
CRS
Column and Row Spaces
CSSE
Column Spaces and Systems of Equations
CSSOC
Column Space Spanned by Original Columns
CSNM
Column Space of a Nonsingular Matrix
RSM
Row Space of a Matrix
CRS
Reading Questions
CRS
Exercises
FS
Four Subsets
LNS
Left Null Space
CCS
Computing Column Spaces
EEF
Extended Echelon Form
FS
Four Subsets
FS
Reading Questions
FS
Exercises
VS
Vector Spaces
VS
Vector Spaces
VS
Vector Spaces
EVS
Examples of Vector Spaces
VSP
Vector Space Properties
RD
Recycling Definitions
VS
Reading Questions
VS
Exercises
S
Subspaces
S
Subspaces
TS
Testing Subspaces
TSS
The Span of a Set
SC
Subspace Constructions
S
Reading Questions
S
Exercises
LISS
Linear Independence and Spanning Sets
LI
Linear Independence
SS
Spanning Sets
VR
Vector Representation
LISS
Reading Questions
LISS
Exercises
B
Bases
B
Bases
BSCV
Bases for Spans of Column Vectors
BNM
Bases and Nonsingular Matrices
OBC
Orthonormal Bases and Coordinates
B
Reading Questions
B
Exercises
D
Dimension
D
Dimension
DVS
Dimension of Vector Spaces
RNM
Rank and Nullity of a Matrix
RNNM
Rank and Nullity of a Nonsingular Matrix
D
Reading Questions
D
Exercises
PD
Properties of Dimension
GT
Goldilocks’ Theorem
RT
Ranks and Transposes
DFS
Dimension of Four Subspaces
PD
Reading Questions
PD
Exercises
D
Determinants
DM
Determinant of a Matrix
EM
Elementary Matrices
DD
Definition of the Determinant
CD
Computing Determinants
DM
Reading Questions
DM
Exercises
PDM
Properties of Determinants of Matrices
DRO
Determinants and Row Operations
DROEM
Determinants, Row Operations, Elementary Matrices
DNMMM
Determinants, Nonsingular Matrices, Matrix Multiplication
PDM
Reading Questions
PDM
Exercises
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
EE
Reading Questions
EE
Exercises
PEE
Properties of Eigenvalues and Eigenvectors
BPE
Basic Properties of Eigenvalues
ME
Multiplicities of Eigenvalues
EHM
Eigenvalues of Hermitian Matrices
PEE
Reading Questions
PEE
Exercises
SD
Similarity and Diagonalization
SM
Similar Matrices
PSM
Properties of Similar Matrices
D
Diagonalization
FS
Fibonacci Sequences
SD
Reading Questions
SD
Exercises
LT
Linear Transformations
LT
Linear Transformations
LT
Linear Transformations
LTC
Linear Transformation Cartoons
MLT
Matrices and Linear Transformations
LTLC
Linear Transformations and Linear Combinations
PI
Preimages
NLTFO
New Linear Transformations From Old
LT
Reading Questions
LT
Exercises
ILT
Injective Linear Transformations
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
ILT
Reading Questions
ILT
Exercises
SLT
Surjective Linear Transformations
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
SLT
Reading Questions
SLT
Exercises
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
IVLT
Reading Questions
IVLT
Exercises
R
Representations
VR
Vector Representations
VR
Vector Representation
CVS
Characterization of Vector Spaces
CP
Coordinatization Principle
VR
Reading Questions
VR
Exercises
MR
Matrix Representations
MR
Matrix Representations
NRFO
New Representations from Old
PMR
Properties of Matrix Representations
IVLT
Invertible Linear Transformations
MR
Reading Questions
MR
Exercises
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
CB
Reading Questions
CB
Exercises
OD
Orthonormal Diagonalization
UTMR
Upper Triangular Matrix Representation
NM
Normal Matrices
OD
Orthonormal Diagonalization
OD
Reading Questions
OD
Exercises
P
Preliminaries
CNO
Complex Number Operations
CNA
Arithmetic with complex numbers
CCN
Conjugates of Complex Numbers
MCN
Modulus of a Complex Number
SET
Sets
SET
Sets
SC
Set Cardinality
SO
Set Operations
Reference
NL
Notation
DL
Definitions
TL
Theorems
EL
Examples
SL
Sage
PT
Proof Techniques
D
Definitions
T
Theorems
L
Language
GS
Getting Started
C
Constructive Proofs
E
Equivalences
N
Negation
CP
Contrapositives
CV
Converses
CD
Contradiction
U
Uniqueness
ME
Multiple Equivalences
PI
Proving Identities
DC
Decompositions
I
Induction
P
Practice
LC
Lemmas and Corollaries
A
Archetypes
A
Archetype A
B
Archetype B
C
Archetype C
D
Archetype D
E
Archetype E
F
Archetype F
G
Archetype G
H
Archetype H
I
Archetype I
J
Archetype J
K
Archetype K
L
Archetype L
M
Archetype M
N
Archetype N
O
Archetype O
P
Archetype P
Q
Archetype Q
R
Archetype R
S
Archetype S
T
Archetype T
U
Archetype U
V
Archetype V
W
Archetype W
X
Archetype X
GFDL
GNU Free Documentation License
Index
NL
Notation
DL
Definitions
TL
Theorems
EL
Examples
SL
Sage
PT
Proof Techniques
A
Archetypes
GFDL
GNU Free Documentation License
Index
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