You're reading from Mathematics of Machine Learning Master linear algebra, calculus, and probability for machine learning

Product type Paperback

Published in May 2025

Publisher Packt

ISBN-13 9781837027873

Length 730 pages

Edition 1st Edition

Concepts

Machine Learning

Author (1):

Tivadar Danka

View More author details

Table of Contents (36) Chapters

Introduction

Part 1: Linear Algebra FREE CHAPTER

1 Vectors and Vector Spaces

2 The Geometric Structure of Vector Spaces

3 Linear Algebra in Practice

4 Linear Transformations

5 Matrices and Equations

6 Eigenvalues and Eigenvectors

7 Matrix Factorizations

8 Matrices and Graphs

References

Part 2: Calculus

9 Functions

10 Numbers, Sequences, and Series

11 Topology, Limits, and Continuity

12 Differentiation

13 Optimization

14 Integration

References

Part 3: Multivariable Calculus

15 Multivariable Functions

16 Derivatives and Gradients

17 Optimization in Multiple Variables

References

Part 4: Probability Theory

18 What is Probability?

19 Random Variables and Distributions

20 The Expected Value

References

Part 5: Appendix

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Index

Appendix A It’s Just Logic

1. Appendix B The Structure of Mathematics

2. Appendix C Basics of Set Theory

3. Appendix D Complex Numbers

2.1 Norms and distances

Previously, we saw that vectors are essentially arrows, starting from the null vector. In addition to their direction, vectors also have magnitude. For example, as we have learned in high school mathematics, the magnitude in the Euclidean plane is defined by

∘ ------- 2 2 ∥x ∥ = x1 + x 2, x = (x1,x2),

while we can calculate the distance between x and y as

∘ --------2-----------2 d(x,y) = (x1 − y1) + (x2 − y2) .

(The function ∥⋅∥ simply denotes the magnitude of a vector.)

Figure 2.2: Magnitude in the Euclidean plane

The magnitude formula ∘ ------- x21 + x22 can be simply generalized to higher dimensions by

∘ ------------ 2 2 n ∥x ∥ = x1 + ⋅⋅⋅+ xn, x = (x1,...,xn) ∈ ℝ .

However, just from looking at this formula, it is not clear why it is defined this way. What does the square root of a sum of squares have to do with distance and magnitude? Behind the scenes, it is just the Pythagorean theorem.

Recall that the Pythagorean theorem states that in right triangles, the squared length of the hypotenuse equals the sum of the squared lengths of the other sides, as illustrated by Figure 2.3.