The Little Book of Linear Algebra

Summary

The Little Book of Linear Algebra A concise, beginner-friendly introduction to the core ideas of linear algebra. Formats Chapter 1. Vectors 1.1 Scalars and Vectors A scalar is a single numerical quantity, most often taken from the real numbers, denoted by $\mathbb{R}$ . Scalars are the fundamental building blocks of arithmetic: they can be added, subtracted, multiplied, and, except in the case of zero, divided. In linear algebra, scalars play the role of coefficients, scaling factors, and entries of larger structures such as vectors and matrices. They provide the weights by which more complex objects are measured and combined. A vector is an ordered collection of scalars, arranged either in a row or a column. When the scalars are real numbers, the vector is said to belong to real $n$ -dimensional space, written $$ \mathbb{R}^n = { (x_1, x_2, \dots, x_n) \mid x_i \in \mathbb{R} }. $$ An element of $\mathbb{R}^n$ is called a vector of dimension $n$ or an n-vector. The number $n$ is called the dimension of the vector space. Thus $\mathbb{R}^2$ is the space of all ordered pairs of real numbers, $\mathbb{R}^3$ the space of all ordered triples, and so on. Example 1.1.1. A 2-dimensional vector: $(3, -1) \in \mathbb{R}^2$ . . A 3-dimensional vector: $(2, 0, 5) \in \mathbb{R}^3$ . . A 1-dimensional vector: $(7) \in \mathbb{R}^1$ , which corresponds to the scalar $7$ itself. Vectors are often written vertically in column form, which emphasizes their role in matrix multiplication: $$ \mathbf{v} = \begin{bmatrix} 2 \ 0 \ 5 \end{bmatrix} \in \mathbb{R}^3. $$ The vertical layout makes the structure clearer when we consider linear combinations or multiply matrices by vectors. Geometric Interpretation In $\mathbb{R}^2$ , a vector $(x_1, x_2)$ can be visualized as an arrow starting at the origin $(0,0)$ and ending at the point $(x_1, x_2)$ . Its length corresponds to the distance from the origin, and its orientation gives a direction in the plane. In $\mathbb{R}^3$ , the same pictur...