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1
Introduction to Linear Algebra
1.1
Applications of Linear Algebra
2
Systems of Linear Equations and Introduction to Matrices
2.1
Systems of Linear Equations
2.1.1
Motivation: Systems of Two Linear Equations In Two Variables
2.1.2
Introduction to Systems of Linear Equations
2.2
Intro to Solving Systems, Gaussian Elimination
2.2.1
Existence and Uniqueness
2.2.2
Triangular Systems, Back-Substitution
2.2.3
Gaussian Elimination, Elementary Operations
2.2.4
Remark on Elementary Operations
2.2.5
Solving Systems of Two Equations in Two Variables
2.3
Introduction to Matrices, Augmented Matrix of a System
2.3.1
Matrices
2.3.2
Augmented Matrix of a System
2.4
Gaussian Elimination and Matrices
2.4.1
Elementary Row Operations
2.4.2
Row Echelon Form
2.4.3
Summary of Row Reduction
2.4.4
Remark on Gaussian Elimination
2.4.5
Systems with Infinitely Many Solutions
2.4.6
Systems with No Solution
2.4.7
Reduced Row Echelon Form
2.4.8
Existence and Uniqueness, Summary
2.4.9
Summary of Solving Systems with Gaussian Elimination
2.4.10
Historical Note
2.4.11
Number of Operations for Row Reduction (Flops)
2.4.12
Computing Note
2.4.13
Exercises
3
Vectors and Geometry
3.1
Introduction to Vectors
3.1.1
Vectors
3.1.2
Vectors in
R
2
3.1.3
Magnitude of a Vector
3.1.4
Unit Vectors
3.1.5
The Standard Basis Vectors
3.1.6
Vectors in
R
3
3.1.7
Vectors in
R
n
3.1.8
Properties of Vector Arithmetic
3.1.9
Linear Combinations of Vectors
3.2
Vector Equations of Lines
3.2.1
Intro to Vector Parametric Form of a Line in the Plane
3.2.2
Lines Using Two Points
3.2.3
Lines in
R
3
3.2.4
Cartesian Equation of a Line in
R
3
3.2.5
Affine Combination Form of a Line
3.3
The Dot Product
3.3.1
Vector Direction
3.3.2
Dot Product of Vectors
3.3.3
Properties of the Dot Product
3.3.4
Angle Between Vectors
3.3.5
Alternate Characterization for the Dot Product
3.3.6
Computing the Angle Between Vectors
3.3.7
Dot Product and Parallel-ness
3.3.8
Dot Product for Vectors in
R
3
3.3.9
Dot Product for Vectors in
R
n
(Inner Product)
3.3.10
Pythagorean Theorem
3.3.11
Angle Between Vectors
3.4
Vector Equations of Planes
3.4.1
Standard Form of a Line, and Normal Vectors
3.4.2
Planes in
R
3
3.4.3
Planes Using Two Vectors: Cross Product
3.4.4
Parametric Equation of a Plane
3.4.5
Planes Using 3 Points
3.4.6
Summary
3.4.7
Planes Parallel to the Coordinate Planes
3.4.8
Intersecting Planes
3.4.9
Parallel Planes
4
Vectors and Linear Equations
4.1
Vector Equations, Span
4.1.1
Vector Equations
4.1.2
Introduction to Span of Vectors in
R
2
,
R
3
4.1.3
Span of
n
Vectors
4.2
Matrix Equations, Matrix-Vector Product
4.2.1
Product of a Matrix with a Vector
4.2.2
Matrix Form of a Linear System
4.2.3
Linear Combination Interpretation
4.2.4
Matrix Equations as Linear Systems
4.2.5
The Identity Matrix
4.2.6
Properties of the Matrix-Vector Product
4.3
Solution Sets of Linear Equations
4.3.1
Homogeneous Systems
4.3.2
Non-homogeneous Systems
4.3.3
Solving a Consistent System in Parametric Vector Form
4.3.4
Intersection of Two Planes
4.3.5
Equation of Line as Intersection of Two Planes
4.3.6
Intersection of Three Planes
5
Matrices and Linear Transformations
5.1
Matrix Operations
5.1.1
Matrix Equality, Addition, Subtraction, and Scalar Multiplication
5.1.2
Properties of Matrix Addition
5.2
Intro to Linear Transformations
5.2.1
Transformations
5.2.2
Linear Transformations
5.3
Matrices and Linear Transformations
5.3.1
Linear Transformations and Matrices
5.3.2
Linear Transformations and Linear Systems
5.4
Linear Transformations in the Plane
5.4.1
Linear Transformations in the Plane
5.4.2
Reflections
5.4.3
Stretching (Contraction/Expansion) and Dilation
5.4.4
Rotation Transformations
5.4.5
General Rotations
5.4.6
Linear Transformations Revisited
5.4.7
Glide Reflection
5.5
Matrix Multiplication
5.5.1
Matrix Multiplication
5.5.2
Matrix Multiplication as a Dot Product (Entry-by-Entry)
5.5.3
Row Perspective of the Matrix-Vector Product and Matrix Multiplication
5.5.4
Properties of Matrix Multiplication
5.5.5
Matrix Multiplication is Not Commutative
5.5.6
The Identity Matrix
5.5.7
Matrix Operations with
1
×
1
Matrices
5.5.8
Matrix Multiplication by Blocks (Partitioned Matrices)
5.6
Elementary Matrices
5.6.1
Elementary Matrices
5.6.2
Permutation Matrices for Row Exchanges
5.7
Inverse of a Matrix
5.7.1
Inverse of a Square Matrix
5.7.2
Matrix Inverses and Linear Transformations
5.7.3
Computing the Inverse of a Matrix (
1
×
1
and
2
×
2
Cases)
5.7.4
Solving Systems with Inverse Matrices
5.7.5
Inverses of Larger Matrices
5.7.6
Matrix Inverses and Elementary Matrices
5.7.7
Inverse of a
3
×
3
Matrix
5.7.8
Properties of Inverse Matrices
5.7.9
Exercises
6
Introduction to Vector Spaces
6.1
Real Coordinate Spaces, Column Space and Null Space
6.1.1
Introduction to Vector Spaces
6.1.2
Subspaces
6.1.3
Spans of Vectors as a Subspace
6.1.4
Vector Spaces of
R
n
6.1.5
Intersection of Subspaces is a Subspace
6.1.6
The Column Space
6.1.7
The Null Space
6.1.8
Contrasting the Column and Null Space
6.2
Linear Independence
6.2.1
Linear Independence
6.3
Basis of a Vector Space
6.3.1
Basis of a Subspace
6.3.2
Basis of the Column Space
6.4
Dimension and Rank
6.4.1
Dimension of a Subspace
6.4.2
Rank of a Matrix
6.4.3
Rank and the Invertible Matrix Theorem
6.5
Coordinate Systems
6.5.1
Coordinate Systems
6.5.2
Change of Basis
7
Determinants
7.1
Introduction to Determinants
7.1.1
Determinants
7.1.2
Determinants of
3
×
3
Matrices
7.1.3
Determinants of Larger Matrices
7.2
Cramer's Rule
7.2.1
The
3
×
3
Case
7.2.2
Cramer's Rule
7.3
Properties of Determinants
7.3.1
Determinants and Row Operations
7.3.2
Evaluating Determinants of Higher Order
7.3.3
Determinants and Invertibility
7.3.4
Number of Operations for Computing Determinants
7.3.5
Exercises
8
General Vector Spaces
8.1
Vector Spaces and Subspaces
8.1.1
Vector Spaces
8.1.2
Subspaces
8.1.3
Subspace Spanned by a Set
8.1.4
Polynomial Spaces
8.1.5
Function Spaces
9
Eigenvalues and Eigenvectors
9.1
Eigenvalues and Eigenvectors
9.1.1
Eigenvalues and Eigenvectors
9.1.2
Determining Eigenvalues and Eigenvectors
9.1.3
The Eigenspace
9.1.4
Computing Eigenvalues
9.1.5
The Characteristic Equation
10
Misc
10.1
Matrix Factorizations, LU Factorziation
10.1.1
LU Factorization
10.1.2
Performing LU Factorization
10.1.3
Algorithm for LU Factorization
10.1.4
Number of Operations for LU Factorization
🔗
Chapter
7
Determinants
🔗
The theory of determinants.
7.1
Introduction to Determinants
7.2
Cramer's Rule
7.3
Properties of Determinants
