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Linear Algebra:
Vectors, Matrices, Spaces, and Transformations
Sam Macdonald
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Front Matter
I
Linear Algebra
1
Matrices and Linear Systems
1.1
Linear Systems and Matrix Equations
Matrix Basics
Encoding Linear Systems
Exercises
1.2
Matrix Arithmetic
Addition and Scalar Multiplication
Matrix Multiplication
Invertibility of Square Matrices
Exercises
1.3
Solving Linear Systems
Gaussian Elimination
Solving Homogeneous Linear Systems
Solving Inhomogeneous Linear Systems
Exercises
2
Introduction to Determinants
2.1
Computing Determinants
2.1
Exercises
2.2
Properties of Determinants
2.2
Exercises
3
Vector Spaces
3.1
Vector Space Basics
3.1
Exercises
3.2
Subspaces, Sums, and Direct Products
3.2
Exercises
4
Span and Bases
4.1
Span
4.1
Exercises
4.2
Linear Independence
4.2
Exercises
4.3
Basis Basics
4.3
Exercises
4.4
Dimension of Vector Spaces
4.4
Exercises
5
Linear Maps
5.1
Linear Map Basics
5.1
Exercises
5.2
Nullspace and Range
Nullspace
Range
Exercises
5.3
The Rank-Nullity Theorem
5.3
Exercises
6
Eigenvectors, Eigenvalues, Diagonalization
6.1
Eigenvectors and Eigenvalues
6.1
Exercises
6.2
The Characteristic Polynomial
6.2
Exercises
6.3
Diagonal Matrices
6.3
Exercises
7
Orthogonality
7.1
Inner Products
7.1
Exercises
7.2
Orthogonal Sets and Projections
7.2
Exercises
7.3
Gram Schmidt and Least Squares
7.3
Exercises
II
Theory of Linear Transformations
1
Determinants: Revisited
2
Inner Product Spaces
3
Change of Bases
4
The Spectral Theorem
5
Extras
5.1
Modules
Backmatter
A
Foundational Knowledge
A.1
Sets, Functions, Constructions
Sets
Functions
Set Constructions
Subsets
Product Sets
Quotient Sets
A.2
Groups, Rings, Fields
B
Notation
C
List of Definitions
D
List of Results
Colophon
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Section
2.1
Computing Determinants
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“”
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Exercises
Exercises
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Computations.
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1
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Coming soon to an OER near you!
Solution
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Formal Proofs.
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2
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Solution
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