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

### Title

Mathematical Analysis 1

(3:3:0)

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### 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):