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

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=== Title ===
 
=== Title ===
Linear and Nonlinear Analysis 1
+
Mathematical Analysis 1
  
 
=== (Credit Hours:Lecture Hours:Lab Hours) ===
 
=== (Credit Hours:Lecture Hours:Lab Hours) ===
(3:3:1)
+
(3:3:0)
  
 
=== Offered ===
 
=== Offered ===
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=== 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 ==
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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.
  
<div style="-moz-column-count:2; column-count:2;">
 
 
#  Abstract Vector Spaces
 
#  Abstract Vector Spaces
 
#* Vector Algebra
 
#* Vector Algebra
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#* Linearly Independent Sets
 
#* Linearly Independent Sets
 
#* Products, Sums, and Complements
 
#* Products, Sums, and Complements
Differentiation rules
+
#  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
+
</div>
+
  
 
=== Textbooks ===
 
=== Textbooks ===

Revision as of 15:10, 4 May 2015

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.

  1. Abstract Vector Spaces
    • Vector Algebra
    • Subspaces and Spans
    • Linearly Independent Sets
    • Products, Sums, and Complements
  2. 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)
  3. 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
  4. Spectral Theory
    • Eigenvalues
    • Diagonalization
    • Special Matrices
    • Positive Definite Matrices
    • The Singular Value Decomposition
    • Generalized Eigenvectors
  5. Metric Space Topology
    • Metric Spaces
    • Properties of Open and Closed Sets
    • Continuous and Uniformly Continuous Mappings
    • Cauchy Sequences and Completeness
    • Compactness
    • Connectedness
  6. Differentiation
    • Directional Derivatives
    • The Derivative
    • The Chain Rule
    • Higher-Order Derivatives
    • The Mean-Value Theorem
    • Taylor’s Theorem
    • Analytic Function Theory (Cauchy Reimann Theorem)
  7. Contraction Mappings and Applications
    • Contraction Mapping Principle
    • Newton’s Method
    • Implicit Function Theorem
    • Inverse Function Theorem
  8. 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):

Additional topics

Courses for which this course is prerequisite