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This document outlines the foundational concepts in algorithms, sets, groups, and linear algebra, essential for understanding mathematical structures and transformations. It details the properties of algorithms, definitions of sets and groups, and distinctions between scalars and vectors. Furthermore, it covers matrix operations, including products, inverses, and transposes, along with coordinate systems and transformations. The content is designed to provide a comprehensive overview for students in mathematical and computational sciences, serving as a groundwork for more advanced topics.
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PSCI 702 Preliminaries September 7, 2005
Table of Content • Algorithms • Sets and Groups • Scalars and Vectors • Matrices • Coordinate Systems • Coordinate Transformations • Operators
Algorithm • “An algorithm is simply a set of finite mathematical operations which, when performed in sequence, lead to the numerical answer to some specific problem.”
Properties of Algorithms: • Finiteness: Algorithm must complete after a finite number of instructions been executed. • Clarity: Each step must be clearly defined, having only one interpretation. • Sequential: Each step has a unique preceding and succeeding step. • Feasibility: All instructions must be able to be performed. Illegal operations are not allowed • Input: 0 or more data values. • Output: 1 or more results.
Sets and Groups • A set is a collection of elements. • Elements are related by some “Law” (‡). • If a, b and c a “set”, where a‡b=c => the set is closed with respect to ‡. • A set with the following properties is called a group: • a‡i=a (unit element) • a-1‡a=I (inverse) • a‡(b‡c)=(a‡b) ‡c (associativity) • ‡ is communitative if a a‡b=b‡a. • ‡ and ^ are distributive if a‡(b^c)=(a‡b)^(a‡c). • Subsets that form a group under addition and scalar multiplication are called fields.
Scalars and Vectors • Scalars are quantities with magnitude. • Vectors are quantities that require more than one number for its specification. Vectors have magnitude and direction. • The number of components that are required for the vector’s specification is called the dimensionality of the vector.
Scalar and Vector Products • Scalar product: • a * b = c • Vector products: • Scalar product or dot product: A.B=c or • Cross product: AxB=C or
Matrices • Matrices are two dimensional vectors with m columns and n rows. • Matrix product is defined as: • AB=C where • Unit Matrix:
Matrices • Inverse: AA-1=1 • Transpose: AT=Aji • Trace of a Matrix: TrA=∑I Aii • Symmetric Matrix: Aij=Aji • Conjugate Transpose: A† • Unitary Matrix: A-1=A† • Normal Matrix: AA†=A†A
Matrices • Determinant is a single parameter that can be used to characterize the matrix.
Coordinate Systems • If the vectors that define the space are locally perpendicular, they are said to form an Orthogonal coordinate frame. • Cartesian (x,y,z) • Cylindrical (ρ,ξ,z) • Polar (r,θ,φ)
Operators • An Operator is a set of instructions represented by a symbol. • Scalar operators such as [d/dx]. • Vector operators such as • Divergence: • Gradient: • Curl:
