Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and here for more.

Vectors - Introduction

Vectors form the basis for many mathematical methods and are important for data analysis. Note that a vector may be seen as a 1-by-n matrix.

Vector We define an ordered set of n equal objects written in a column vector of order n, and the row counterpart of m objects a row vector (of order m). Please keep in mind that these definitions are simplified and cover only part of the exact, mathematical definition. However, the definition given here is sufficient for our purposes concerning data analysis.

To denote a specific vector, we shall use a lowercase, bold letter, such as a, for example. Whether this vector a is a column or row vector, will usually be clear from the context in which the letter is used. When written explicitly, vectors are put in parenthesis.

Scalar The n elements that form a vector according to definition above are called scalars. All scalars are taken from the same basic set. For most purposes the basic set is the space of real numbers R.

To denote a certain element of a vector, we shall use a lowercase letter (to specify the vector of which we are taking an element) with an index (for the index within the vector), such as a2, for example.

Example: The vector v:=(4, 3, 5.1, π, -e) is a row vector of order 5, which we also could denote by (vk). v5 is the fifth element of this vector, and thus -e.

It is an important aspect of vector and matrix algebra that we have two sets of objects to handle: the vectors (or matrices, respectively) as an entity and the scalars contained in these entities (however, matrices can also be seen as vectors of vectors of scalars).