# Difference between revisions of "Math 344: Mathematical Analysis 1"

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=== Title === | === Title === | ||

− | + | Mathematical Analysis 1 | |

=== (Credit Hours:Lecture Hours:Lab Hours) === | === (Credit Hours:Lecture Hours:Lab Hours) === | ||

− | (3:3: | + | (3:3:0) |

=== Offered === | === Offered === | ||

Line 14: | Line 14: | ||

=== Description === | === Description === | ||

+ | Development of the theory of vector spaces, linear maps, inner product spaces, spectral theory, metric space topology, differentiation, contraction mappings and convex analysis. | ||

== Desired Learning Outcomes == | == Desired Learning Outcomes == | ||

Line 23: | Line 24: | ||

Students will have a solid understanding of the concepts listed below. They will be able to prove theorems that are central to this material, including theorems that they have not seen before. They will understand connections between the concepts taught, and be able to relate them to other mathematical material that they have studied. They will be able to perform the related computations on small, simple problems. | Students will have a solid understanding of the concepts listed below. They will be able to prove theorems that are central to this material, including theorems that they have not seen before. They will understand connections between the concepts taught, and be able to relate them to other mathematical material that they have studied. They will be able to perform the related computations on small, simple problems. | ||

− | |||

# Abstract Vector Spaces | # Abstract Vector Spaces | ||

#* Vector Algebra | #* Vector Algebra | ||

Line 29: | Line 29: | ||

#* Linearly Independent Sets | #* Linearly Independent Sets | ||

#* Products, Sums, and Complements | #* Products, Sums, and Complements | ||

− | # | + | # Linear Transformations and Linear Systems |

− | Linear Transformations and Linear Systems | + | #* Definitions and Examples |

− | Definitions and Examples | + | #* Rank, Range, and Nullity |

− | Rank, Range, and Nullity | + | #* Matrix Representations |

− | Matrix Representations | + | #* Change of Basis and Similarity |

− | Change of Basis and Similarity | + | #* Invariant Subspaces |

− | Invariant Subspaces | + | #* Linear Systems (Elementary Matrices, Row Echelon Form, Inverses, Determinants) |

− | Linear Systems (Elementary Matrices, Row Echelon Form, Inverses, Determinants) | + | # Inner Product Spaces |

− | Inner Product Spaces | + | #* Definitions and Examples |

− | Definitions and Examples | + | #* Orthonormal Sets |

− | Orthonormal Sets | + | #* Gram Schmidt Orthogonalization |

− | Gram Schmidt Orthogonalization | + | #* Normed Linear Spaces |

− | Normed Linear Spaces | + | #* The Adjoint of a Linear Transformation |

− | The Adjoint of a Linear Transformation | + | #* Fundamental Subspaces |

− | Fundamental Subspaces | + | #* Projectors |

− | Projectors | + | #* Least Squares |

− | Least Squares | + | # Spectral Theory |

− | Spectral Theory | + | #* Eigenvalues |

− | Eigenvalues | + | #* Diagonalization |

− | Diagonalization | + | #* Special Matrices |

− | Special Matrices | + | #* Positive Definite Matrices |

− | Positive Definite Matrices | + | #* The Singular Value Decomposition |

− | The Singular Value Decomposition | + | #* Generalized Eigenvectors |

− | Generalized Eigenvectors | + | # Metric Space Topology |

− | Metric Space Topology | + | #* Metric Spaces |

− | Metric Spaces | + | #* Properties of Open and Closed Sets |

− | Properties of Open and Closed Sets | + | #* Continuous and Uniformly Continuous Mappings |

− | Continuous and Uniformly Continuous Mappings | + | #* Cauchy Sequences and Completeness |

− | Cauchy Sequences and Completeness | + | #* Compactness |

− | Compactness | + | #* Connectedness |

− | Connectedness | + | # Differentiation |

− | Differentiation | + | #* Directional Derivatives |

− | Directional Derivatives | + | #* The Derivative |

− | The Derivative | + | #* The Chain Rule |

− | The Chain Rule | + | #* Higher-Order Derivatives |

− | Higher-Order Derivatives | + | #* The Mean-Value Theorem |

− | The Mean-Value Theorem | + | #* Taylor’s Theorem |

− | Taylor’s Theorem | + | #* Analytic Function Theory (Cauchy Reimann Theorem) |

− | Analytic Function Theory (Cauchy Reimann Theorem) | + | # Contraction Mappings and Applications |

− | Contraction Mappings and Applications | + | #* Contraction Mapping Principle |

− | Contraction Mapping Principle | + | #* Newton’s Method |

− | Newton’s Method | + | #* Implicit Function Theorem |

− | Implicit Function Theorem | + | #* Inverse Function Theorem |

− | Inverse Function Theorem | + | # Convex Analysis |

− | Convex Analysis | + | #* Convex Sets |

− | Convex Sets | + | #* Convex Combinations |

− | Convex Combinations | + | #* Examples (Hyperplanes, Halfspaces, Cones) |

− | Examples (Hyperplanes, Halfspaces, Cones) | + | #* Geometry of Convex Sets (separation theorem) |

− | Geometry of Convex Sets (separation theorem) | + | #* Convex Functions |

− | Convex Functions | + | #* Graphs and Epigraphs |

− | Graphs and Epigraphs | + | #* Inequalities |

− | Inequalities | + | #* The Convex Conjugate |

− | The Convex Conjugate | + | |

− | + | ||

=== Textbooks === | === Textbooks === |

## Revision as of 14:10, 4 May 2015

## Contents

## Catalog Information

### Title

Mathematical Analysis 1

### (Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

### Offered

F

### Prerequisite

Math 290, Math 313, Math 314; concurrent with Math 334, Math 345

### Description

Development of the theory of vector spaces, linear maps, inner product spaces, spectral theory, metric space topology, differentiation, contraction mappings and convex analysis.

## Desired Learning Outcomes

### Prerequisites

Math 290, Math 313, Math 314; concurrent with Math 334, Math 345

### Minimal learning outcomes

Students will have a solid understanding of the concepts listed below. They will be able to prove theorems that are central to this material, including theorems that they have not seen before. They will understand connections between the concepts taught, and be able to relate them to other mathematical material that they have studied. They will be able to perform the related computations on small, simple problems.

- Abstract Vector Spaces
- Vector Algebra
- Subspaces and Spans
- Linearly Independent Sets
- Products, Sums, and Complements

- Linear Transformations and Linear Systems
- Definitions and Examples
- Rank, Range, and Nullity
- Matrix Representations
- Change of Basis and Similarity
- Invariant Subspaces
- Linear Systems (Elementary Matrices, Row Echelon Form, Inverses, Determinants)

- Inner Product Spaces
- Definitions and Examples
- Orthonormal Sets
- Gram Schmidt Orthogonalization
- Normed Linear Spaces
- The Adjoint of a Linear Transformation
- Fundamental Subspaces
- Projectors
- Least Squares

- Spectral Theory
- Eigenvalues
- Diagonalization
- Special Matrices
- Positive Definite Matrices
- The Singular Value Decomposition
- Generalized Eigenvectors

- Metric Space Topology
- Metric Spaces
- Properties of Open and Closed Sets
- Continuous and Uniformly Continuous Mappings
- Cauchy Sequences and Completeness
- Compactness
- Connectedness

- Differentiation
- Directional Derivatives
- The Derivative
- The Chain Rule
- Higher-Order Derivatives
- The Mean-Value Theorem
- Taylor’s Theorem
- Analytic Function Theory (Cauchy Reimann Theorem)

- Contraction Mappings and Applications
- Contraction Mapping Principle
- Newton’s Method
- Implicit Function Theorem
- Inverse Function Theorem

- Convex Analysis
- Convex Sets
- Convex Combinations
- Examples (Hyperplanes, Halfspaces, Cones)
- Geometry of Convex Sets (separation theorem)
- Convex Functions
- Graphs and Epigraphs
- Inequalities
- The Convex Conjugate

### Textbooks

Possible textbooks for this course include (but are not limited to):