# Linear Algebra Topics

## 1. Systems of Linear Equations and Matrices

###### 1.1. Systems of Linear Equations

- Linear Equations
- Systems of Linear Equations
- Solutions to Systems of 2-Variables and 3-Variables
- Homogenous Linear Systems

###### 1.2. Introduction to Matrices and Elementary Row Operations

- Matrices and Elementary Row Operations
- Row Echelon Form of a Matrix (REF)
- Gaussian Elimination and Back Substitution
- Reduced Row Echelon Form of a Matrix (RREF)
- Gauss-Jordan Elimination

###### 1.3. Operations on Matrices, and Special Types of Matrices

- Matrix Addition and Subtraction
- Scalar Multiples of Matrices
- The Transpose of a Matrix
- The Trace of a Square Matrix
- Matrix Multiplication
- Zero Matrices
- Diagonal Matrices
- Triangular Matrices
- Symmetric Matrices
- Identity Matrices
- Inverse of a Matrix
- Inverse of Diagonal Matrices
- Powers of a Matrix
- Matrix Polynomials
- Elementary Matrices
- Finding a Matrix's Inverse with Elementary Matrices
- Finding a Matrix's Inverse with Row Operations

## 2. Determinants

###### 2.1. The Determinant of Square Matrices

- Evaluating 2x2 Determinants and Matrix Inverses
- Minor and Cofactor Entries
- Evaluating nxn Determinants with Minor and Cofactor Entries of a Matrix
- Determinants of Triangular and Diagonal Matrices
- Effects of Elementary Matrix Operations on Determinants
- Determinants of Elementary Matrices
- Evaluating Determinants by Row Reduction
- Combinatorial Approach to Determinants
- Properties of Determinants
- Adjoint Matrices
- Finding a Matrix Inverse with its Determinant and Adjoint
- Cramer's Rule for Systems of Linear Equations

## 3. Vector Geometry

###### 3.1. Vectors in Euclidean Space

- Scalars and Vectors
- Euclidean Space
- Basic Properties of Vectors
- Determining a Vector Given Two Points
- Translation Equations
- The Norm of a Vector
- Unit Vectors
- Vector Dot Product (Euclidean Inner Product)
- Orthogonal (Perpendicular) Vectors
- Orthogonal Projections
- Vector Cross Product
- Lagrange's Identity
- Standard Unit Vectors

###### 3.2. Vector Geometry in Euclidean Space

- The Areas of Parallelograms and Triangles in 3-Space
- The Area of a Parallelogram in 2-Space
- Scalar Triple Products
- The Volume of a Parallelepiped in 3-Space
- Point-Normal Form of a Plane
- Vector Form Equations of a Plane
- Parametric Equations of Lines
- Lines of Intersection Between Two Planes
- The Distance Between a Plane and a Point
- The Distance Between Parallel Planes
- The Distance Between Two Vectors

###### 3.3. Transformations in Euclidean Space

## 4. Abstract Fields

###### 4.1. Fields

## 5. Vector Spaces

###### 5.1. Introduction to Vector Spaces

- Vector Spaces
- The Vector Space of n-Component Vectors
- The Vector Space of m x n Matrices
- The Vector Space of Lines Through the Origin of R2
- The Zero Vector Space
- Further Examples of Vector Spaces
- Properties of Vector Spaces
- Determining Whether a Set is a Vector Space
- Vector Subspaces ( Examples 1 | Examples 2 )
- The Vector Subspace of 2 x 2 Matrices
- The Vector Subspace of Real-Valued Continuous Functions
- Vector Subspaces of Homogenous Systems of Rn
- Vector Subspace Sums ( Examples 1 | Examples 2 )
- Vector Sum Theorems
- Direct Sum Theorems
- The Intersection and Union of Subspaces
- Solution Spaces of Homogenous Linear Systems
- Vector Spaces Review

## 6. Linear Independence, Spanning Sets, and Bases

###### 6.1. Linear Independence and Spanning Sets

- Linear Combinations of a Set of Vectors
- Span of a Set of Vectors
- Spanning Set of a Vector Space ( Examples 1 )
- Finite and Infinite-Dimensional Vector Spaces ( Examples 1 )
- Linear Independence and Dependence ( Examples 1 | Examples 2 | Examples 3 | Examples 4 | Examples 5 )
- Theorems Regarding Linear Independence and Dependence
- Linear Dependence Lemma
- Finite-Dimensional Linearly Independent Set of Vectors Theorem
- Infinite/Finite-Dimensional Vector Space Comparison Theorem
- Infinite-Dimensional Vector Space Theorem
- Linear Independence and Spanning Sets Review

###### 6.2. Bases and the Dimension of Vector Spaces

- Basis of a Vector Space ( Examples 1 | Examples 2 | Examples 3 )
- Theorems Regarding a Basis of a Vector Space
- Finding a Basis for a Set of Vectors
- Dimension of a Vector Space ( Examples 1 )
- Sufficient Conditions for a Set of Vectors to be a Basis
- The Dimension of a Sum of Subspaces ( Examples 1 | Examples 2 )
- The Dimension of a Direct Sum of Subspaces
- Bases and Dimension Review

## 7. Linear Maps

###### 7.1. Linear Maps

- Linear Maps ( Examples 1 | Examples 2 | Examples 3 | Examples 4 )
- Linear Maps Defined by Bases
- Linear Maps from Fn to Fm
- Addition, Multiples, and Compositions of Linear Maps ( Examples 1 )
- Null Space of a Linear Map
- Range of a Linear Map ( Examples 1 )
- Injective and Surjective Linear Maps ( Examples 1 | Examples 2 | Examples 3 | Examples 4 )
- The Dimension of The Null Space and Range ( Examples 1 | Examples 2 | Examples 3 )
- Linear Maps Review

###### 7.2. Matrices of Linear Maps

## 8. Polynomials

###### 8.1. Polynomials

- Properties of Polynomials ( Examples 1 | Examples 2 )
- The Fundamental Theorem of Algebra
- The Factorization of Polynomials with Complex Coefficients
- Pairs of Complex Roots for Polynomials with Real Coefficients
- The Factorization of Polynomials with Real Coefficients
- Root-Finding Techniques
- Polynomials Review

## 9. Eigenvalues of Linear Operators

###### 9.1. Eigenvalues

- Invariant Subspaces ( Examples 1 | Examples 2 )
- Eigenvalues and Eigenvectors ( Examples 1 | Examples 2 | Examples 3 | Examples 4 | Examples 5 | Examples 6 )
- Polynomials Applied to Linear Operators ( Examples 1 | Examples 2 )
- The Existence of an Eigenvalue on Finite-Dimensional Complex Vector Spaces
- Upper Triangular Matrices of Linear Operators
- Upper Triangular Matrices for Operators on Complex Vector Spaces
- Determining Eigenvalues from Upper Triangular Matrices of Linear Operators
- Diagonal Matrices of Linear Operators ( Examples 1 | Examples 2 | Examples 3 )
- Invariant Subspaces in Finite-Dimensional Real Vector Spaces
- Projection Operators
- The Existence of an Eigenvalue on an Odd-Dimensional Real Vector Space
- Eigenvalues Review

## 10. Inner Product Spaces

###### 10.1. Inner Product Spaces

- Inner Product Spaces ( Examples 1 )
- Formulas for The Inner Product
- The Pythagorean Theorem for Inner Product Spaces ( Examples 1 )
- The Cauchy-Schwarz Inequality ( Examples 1 )
- The Triangle Inequality for Inner Product Spaces
- The Parallelogram Identity for Inner Product Spaces
- Inner Product Spaces Review

###### 10.2. Orthonormal and Orthogonal Spaces

## 11. Linear Functionals and Adjoint Linear Operators

###### 11.1. Linear Functionals and Adjoint Linear Operators

- Linear Functionals ( Examples 1 )
- The Adjoint of a Linear Map
- Properties of Adjoints of Linear Maps
- Determining the Adjoint of a Linear Map
- The Null Space and Range of the Adjoint of a Linear Map
- Injectivity and Surjectivity of the Adjoint of a Linear Map
- Eigenvalues of the Adjoint of a Linear Map
- The Conjugate Transpose of a Matrix
- The Matrix of the Adjoint of a Linear Map
- Self-Adjoint Linear Operators
- Eigenvalues of Self-Adjoint Linear Operators
- Self-Adjoint Linear Operators over Complex Vector Spaces
- Normal Linear Operators

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###### References

- 1. Elementary Linear Algebra and Applications (11th Edition) by Howard Anton and Chris Rorres

- 2. Linear Algebra Done Right (2nd Edition) by Sheldon Axler