An Introduction to Optimization, 4th Edition

CHAPTER 2

VECTOR SPACES AND MATRICES

2.1 Vector and Matrix

We define a column n-vector to be an array of n numbers, denoted

The number a_i is called the ith component of the vector a. Denote by the set of real numbers and by ⁿ the set of column n-vectors with real components. We call ⁿ an n-dimensional real vector space. We commonly denote elements of ⁿ by lowercase bold letters (e.g., x). The components of x ⁿ are denoted x₁, …, x_n.

We define a row n-vector as

The transpose of a given column vector a is a row vector with corresponding elements, denoted a. For example, if

then

Equivalently, ...