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Linear algebra: the theory of vector spaces and linear transformations
Aaron Greicius
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Front Matter
Colophon
Acknowledgements
0
Foundations
0.1
Sets
0.1
Exercises
0.2
Functions
0.2
Exercises
0.3
Tuples and Cartesian products
0.4
Logic
0.4.1
Propositional logic
0.4.2
Predicate logic
0.4.3
Exercises
0.5
Proof techniques
0.5.1
Logical structure
Implication
Disjunction
Equivalence
Chains of implications/equivalences
Proof by contradiction
0.5.2
Equalities
Chain of equalities
0.5.3
Basic set properties
Set inclusion
Set equality
0.5.4
Basic function properties
Function equality
Injective, surjective, bijective
0.5.5
Mathematical induction
Proof by induction
Proof by strong induction
0.6
Complex numbers
0.6.1
Definition of
C
0.6.2
Absolute value and complex conjugation
0.6.3
Exercises
0.7
Polynomials
0.7.1
Basic definitions
0.7.2
Degree of a polynomial
0.7.3
Factoring polynomials
0.7.4
Exercises
1
Systems of linear equations
1.1
Systems of linear equations
1.1.1
Systems of linear equations
1.1.2
Row operations
1.1.3
Exercises
1.2
Gaussian elimination
1.2.1
Row echelon matrices
1.2.2
Gaussian elimination
Model example
1.2.3
Exercises
1.3
Solving linear systems
1.3.1
Model example continued
1.3.2
General method for solving linear systems
1.3.3
Exercises
2
Matrices, their arithmetic, and their algebra
2.1
Matrix arithmetic
2.1.1
The basics
2.1.2
Addition, subtraction and scalar multiplication
2.1.3
Matrix multiplication
2.1.4
Alternative methods of multiplication
2.1.5
Transpose of a matrix
2.1.6
Exercises
2.2
Matrix algebra
2.2
Exercises
2.3
Invertible matrices
2.3.1
Invertible matrices
2.3.2
Powers of matrices, matrix polynomials
2.3.3
Exercises
2.4
The invertibility theorem
2.4.1
Elementary matrices
2.4.2
Interlude on matrix equations
2.4.3
The invertibility theorem
2.4.4
Invertibility algorithms
2.4.5
Some theoretical loose ends
2.4.6
Exercises
2.5
The determinant
2.5.1
Definition of the determinant
2.5.2
Expansion along rows and columns
2.5.3
Row operations and determinant
2.5.4
Exercises
3
Vector spaces and linear transformations
3.1
Real vector spaces
3.1.1
Definition of a vector space
3.1.2
Examples
3.1.3
General properties
3.1.4
Exercises
3.2
Linear transformations
3.2.1
Linear transformations
3.2.2
Matrix transformations
3.2.3
Rotations, reflections, and orthogonal projections
3.2.4
Additional examples
3.2.5
Composition of linear transformations and matrix multiplication
3.2.6
Exercises
3.3
Subspaces
3.3.1
Definition of subspace
3.3.2
Subspaces of
R
n
3.3.3
Important subspaces of
M
n
n
3.3.4
Important subspaces of
F
(
I
,
R
)
3.3.5
Exercises
3.4
Null space and image
3.4.1
Null space and image of a linear transformation
3.4.2
Injective and surjective linear transformations
3.4.3
Exercises
3.5
Span and linear independence
3.5.1
Span
3.5.2
Linear independence
3.5.3
Linear independence in function spaces
3.5.4
Exercises
3.6
Bases
3.6.1
Bases of vector spaces
3.6.2
Bases and linear transformations
3.6.3
Exercises
3.7
Dimension
3.7.1
Dimension of a vector space
3.7.2
Street smarts!
3.8
Rank-nullity theorem and fundamental spaces
3.8.1
The rank-nullity theorem
3.8.2
Fundamental spaces of matrices
3.8.3
Contracting and expanding to bases
3.8.4
Exercises
3.9
Isomorphisms
3.9.1
Isomorphisms
4
Eigenvectors and diagonalization
4.1
Coordinate vectors
4.1.1
Coordinate vectors
4.1.2
Coordinate vector transformation
4.1.3
Exercises
4.2
Matrix representations of linear transformations
4.2.1
Matrix representations of linear transformations
4.2.2
Matrix representations as models
4.2.3
Choice of basis
4.2.4
Exercises
4.3
Change of basis
4.3.1
Change of basis matrix
4.3.2
Change of basis for transformations
4.3.3
Similarity and the holy commutative tent of linear algebra
4.3.4
Exercises
4.4
Eigenvectors and eigenvalues
4.4.1
Eigenvectors
4.4.2
Finding eigenvalues and eigenvectors systematically
4.4.3
Properties of the characteristic polynomial
4.4.4
Exercises
4.5
Diagonalization
4.5.1
Diagonalizable transformations
4.5.2
Linear independence of eigenvectors
4.5.3
Diagonalizable matrices
4.5.4
Algebraic and geometric multiplicity
4.5.5
Exercises
5
Inner product spaces
5.1
Inner product spaces
5.1.1
Inner products
5.1.2
Norm and distance
5.1.3
Cauchy-Schwarz inequality, triangle inequalities, and angles between vectors
5.1.4
Choosing your inner product
5.1.5
Exercises
5.2
Orthogonal bases
5.2.1
Orthogonal vectors and sets
5.2.2
Orthogonal bases
5.2.3
Coordinate vectors and matrix representations
5.2.4
Exercises
5.3
Orthogonal projection
5.3.1
Orthogonal complement
5.3.2
Orthogonal Projection
5.3.3
Orthogonal projection in
R
2
and
R
3
5.3.4
Trigonometric polynomial approximation
5.3.5
Least-squares solution to linear systems
5.3.6
Exercises
5.4
The spectral theorem
5.4.1
Self-adjoint operators
5.4.2
The spectral theorem for self-adjoint operators
5.4.3
Exercises
Back Matter
A
Notation
B
Exercises
C
Definitions
D
Theory and procedures
E
Examples
F
Sage examples
G
Figures and video examples
H
Mantras and fiats
Index
🔗
A
Notation
B
Exercises
C
Definitions
D
Theory and procedures
E
Examples
F
Sage examples
G
Figures and video examples
H
Mantras and fiats
Index
