In this section, we will explore the concepts of vector spaces and subspaces. These are very important to our understanding of linear algebra. In fact, if we do not have an understanding of vector spaces and subspaces, we do not truly have an understanding of how to solve linear algebra problems.
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Table Of Contents
Hands-On Mathematics for Deep Learning
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Hands-On Mathematics for Deep Learning
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Overview of this book
Most programmers and data scientists struggle with mathematics, having either overlooked or forgotten core mathematical concepts. This book uses Python libraries to help you understand the math required to build deep learning (DL) models.
You'll begin by learning about core mathematical and modern computational techniques used to design and implement DL algorithms. This book will cover essential topics, such as linear algebra, eigenvalues and eigenvectors, the singular value decomposition concept, and gradient algorithms, to help you understand how to train deep neural networks. Later chapters focus on important neural networks, such as the linear neural network and multilayer perceptrons, with a primary focus on helping you learn how each model works. As you advance, you will delve into the math used for regularization, multi-layered DL, forward propagation, optimization, and backpropagation techniques to understand what it takes to build full-fledged DL models. Finally, you’ll explore CNN, recurrent neural network (RNN), and GAN models and their application.
By the end of this book, you'll have built a strong foundation in neural networks and DL mathematical concepts, which will help you to confidently research and build custom models in DL.
Table of Contents (19 chapters)
Preface
Section 1: Essential Mathematics for Deep Learning
Free Chapter
Linear Algebra
Vector Calculus
Probability and Statistics
Optimization
Graph Theory
Section 2: Essential Neural Networks
Linear Neural Networks
Feedforward Neural Networks
Regularization
Convolutional Neural Networks
Recurrent Neural Networks
Section 3: Advanced Deep Learning Concepts Simplified
Attention Mechanisms
Generative Models
Transfer and Meta Learning
Geometric Deep Learning
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, then 
. 
is a straight line, 
is all the possible points in the xy-plane, and 
is all the possible points in the xyz-plane—that is, 3-dimensional space, and so on.
for all 
.
, there exists an additive inverse such that 
.
, there exists a multiplicative identity such that 
.
, 
.
.
and 
for all 
and for all 
.
, which implies that 
.
is the set of all linear combinations that can be made using the n vectors. Therefore, 
if the vectors are linearly independent and span V completely; then, the vectors 
are the basis of V.
. Then, S can only be a subspace if it follows the three rules, stated as follows:
and 
, which implies that S is closed under addition
and 
so that 
, which implies that S is closed under scalar multiplication
, then their sum is 
, where the result is also a subspace of V.
is as follows:
