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Free energy, G, is an expression of the energy available to do work—for exam- ple, the work of driving chemical reactions

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we can think of the columns of A as sp ecifying diﬀerent directions we can tra v el from the origin (the p oin t sp eciﬁed b y the v ector of all zeros), and determine ho w many wa ys there are of reac hing b . In this view, each element of x <span>sp eciﬁes ho w far we should trav el in eac h of these directions, with x i sp ecifying how far to mo v e in the direction of column :<span><body><html>

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chapter will completely omit many important linear algebra topics that are not essential for understanding deep learning. 2.1 Scalars, Vectors, Matrices and Tensors The study of linear algebra involves several types of mathematical objects: • <span>Scalars : A scalar is just a single number, in contrast to most of the other objects studied in linear algebra, which are usually arrays of multiple numbers. We write scalars in italics. We usually give scalars lower-case variable names. When we introduce them, we specify what kind of number they are. For 31 CHAPTER 2. LINEAR ALGEBRA example, we might say “Let s ∈ R be the slope of the line,” while deﬁning a real-valued scalar, or “Let n ∈ N be the number of units,” while deﬁning a natural number scalar. • Vectors : A vector is an array of numbers. The numbers are arranged in order. We can identify each individual number by its index in that ordering. Typically we give vectors lower ca