Math C: Applied Mathematics

The applied-mathematics module covers linear algebra, calculus, optimization, numerical stability, and information theory. Beyond the formula, check scale, conditioning, error, and the algorithm's stopping rule.

Prerequisites

• Be familiar with high-school algebra, functions, exponents, and logarithms. Linear algebra and calculus begin with their basic objects and rates of change.
• Entropy and coding need basic probability from 8.1 Probability Space and Conditional Probability: First Build the Model, Then Update Your Judgment. Numerical examples require running Python and installing the dependencies listed on each page.

Coverage

Chapter 10 covers vectors, matrices, systems of equations, and approximation. Chapter 11 connects derivatives, integrals, and optimization. Chapter 12 examines information, floating-point rounding, conditioning, and algorithmic stability.

Chapter 10

• 10.1 Vectors and Inner Products: Turning Objects into Comparable Coordinates
• 10.2 Matrices, Linear Transformations, and Systems of Equations: From Structure to Solvability
• 10.3 Least Squares, Eigen Decomposition, and SVD: From Exact Solutions to Best Approximations

Chapter 11

• 11.1 Limits, Derivatives, and the Chain Rule: Describing Local Change
• 11.2 Integrals and Numerical Integration: From Local Rates to Cumulative Totals
• 11.3 Gradients and Optimization: From Descent Direction to Verifiable Solutions

Chapter 12

• 12.1 Self-Information, Entropy, and Coding: How Uncertainty Becomes Bits
• 12.2 Floating-Point Numbers and Rounding: How Finite Bits Approximate Real Numbers
• 12.3 Condition Numbers and Numerical Stability: Preventing Error Amplification

Outcomes

• Express comparison, transformation, and fitting problems with vectors and matrices, checking dimensions and scale.
• Apply derivatives, integrals, and gradients to rates, totals, and optimization, and explain the stopping rule.
• Distinguish sensitivity in a problem from error introduced by an algorithm, and assess whether the computed result is trustworthy.

How to use this map

For machine learning preparation, begin with Chapters 10 and 11. Return to Chapter 12 when numerical results misbehave. Estimate the scale of an answer before running each example. A screen full of decimal places shows that the formatter worked hard; accuracy needs separate evidence.