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

### Title

Linear and Nonlinear Analysis 1

(3:3:1)

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### Prerequisite

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

## 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. Differentiation rules

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):