Rectangular Arrays
Introduction
A rectangular array, also known as a matrix, is a fundamental concept in mathematics, particularly in the fields of linear algebra, computer science, and physics. It is a set of numbers arranged in rows and columns, forming a rectangle. The numbers within the array are referred to as elements.
Definition
A rectangular array is defined as a collection of 'm' rows and 'n' columns, where 'm' and 'n' are positive integers. Each element in the array is identified by two indices, one representing its row and the other its column. The size of a rectangular array is determined by the number of rows and columns it contains, typically expressed as 'm x n'.
Properties
Rectangular arrays have several properties that make them useful in various mathematical and scientific disciplines:
- Order: The order of a rectangular array is given by the number of rows and columns it contains. For example, a 3x4 array has 3 rows and 4 columns.
- Elements: The individual items in a rectangular array are called elements. Each element is identified by its row and column indices.
- Row and Column Vectors: A row in a rectangular array is known as a row vector, while a column is known as a column vector.
- Square Array: If the number of rows equals the number of columns, the array is called a square array.
- Zero Array: An array in which all elements are zero is called a zero array.
Applications
Rectangular arrays have wide applications in various fields:
- Linear Algebra: In linear algebra, rectangular arrays are used to represent linear equations and transformations.
- Computer Science: In computer science, rectangular arrays, or matrices, are used in graphics transformations, algorithms, and data structures.
- Physics: In physics, rectangular arrays are used to represent physical quantities such as the state of a quantum system or the transformation of coordinates in space.
Operations on Rectangular Arrays
There are several operations that can be performed on rectangular arrays, including addition, subtraction, multiplication, and division. These operations follow specific rules and are fundamental to many mathematical procedures.