Definition and Representation of Vectors in Matrix Form

In mathematics, vectors are fundamental entities that are widely used in various fields such as physics, engineering, and computer science. The concept of vectors can be defined in different contexts, depending on the space in which they are considered. This article explores the definition and representation of vectors in matrix form, focusing on vector algebra, general abstract vector spaces, and Euclidean spaces.

Vector Algebra and Free Vectors

In vector algebra, a foundational branch of mathematics, vectors are defined as equivalence classes of bound vectors, or fixed vectors. A bound vector AB, represented as a directed line segment from point A to point B, is considered in Euclidean spaces. Two bound vectors AB and CD are considered equal if they share the same direction, including the sense (or orientation) and the same length. This implies that a free vector, which has no specified starting point, is not defined in any matrix form within vector algebra.

General Abstract Vector Spaces

In a more abstract setting, general abstract vector spaces are defined by their ten axioms: L1 to L10. These axioms form the basis for operations within such spaces. In these spaces, vectors are elements of V, and matrices do not directly play a role in defining vectors or their operations.

Euclidean Spaces and Matrix Representation

In Euclidean n-dimensional spaces, such as Rsup">n or Fsup">n (where F equals a general numerical field), vectors are ordered n-tuples of scalar numbers. Vectors can be represented in three equivalent ways:

X x1 x2 ... xn X [x1 x2 ... xn], a row vector of scalars X [x1 x2 ... xn]T, a column vector of scalars

The third representation is the most convenient in both theoretical formulas and practical applications. This is because a vector in matrix form is either a 1-by-n matrix or an n-by-1 matrix, depending on whether it is written as a row or a column vector. In a linear system of equations, this form is particularly useful. Matrices A of size m x n are used to represent the coefficient matrix of the system, while X is the solution vector, and b is the constant vector.

Matrix Form of Linear Systems

A linear system can be written in matrix form as:

AX b, for a non-homogeneous system AX 0, for a homogeneous system

Here, A ∈ Mmn, X ∈ Rn, and b ∈ Rm. The solution set of the system, denoted by S, is defined as:

S {X ∈ Rn : A X b}

The nature of the solution set (whether it is consistent or not) and the determination of S when S is not an empty set can be analyzed using the augmented matrix of the system, denoted as [ A | b ]. For homogeneous systems, the augmented matrix is not relevant since they are always consistent.

Further Properties and References

For a deeper understanding of the properties of Euclidean n-dimensional spaces and linear systems, readers can consult standard textbooks, monographs, or lecture notes on linear algebra and matrix theory. Some recommended resources include:

Alexandru Carausu, Linear Algebra - Theory and Applications, Matrix Rom Publishers, Bucharest, 1999.

These resources provide comprehensive insights into the properties of vectors and matrices in different contexts, expanding the reader's knowledge of vector spaces and linear algebra.