MA5242 Wavelets Lecture 1 Numbers and Vector Spaces Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg Tel (65) 6874-2749

Numbers integers, is a ring rationals, integers modulo a prime, are fields reals, is a complete field under the topology induced by the absolute value complex numbers, is a complete field under the topology induced by the absolute value that is algebraically closed (every polynomial with coefficients in C has a root in C )

Polar Representation of Complex Numbers Z – integer, R-real, Q-rational, C-complex Polar Representation of the Field Euler’s Formula Cartesian Geometry

Problem Set 1 1. State the definition of group, ring, field. 2. Give addition & multiplication tables for Determine which are fields? 3. What is a Cauchy Sequence? Why is Q not complete and why is R complete. 4. Show that R is not algebraically closed. 5. Derive the following:

Vector Spaces over a Field Definition: V is a vector space over a field F if V is an abelian (commutative) group under addition and, for every this means that is a homomorphism of V into V, and, for every this means that is a ring homomorphism Convention:

Examples of Vector Spaces a positive integer, a field with operations is a vector space over the field

Examples of Vector Spaces Example 1. The set of functions f : R  R below Example 2. The set of f : R  R such that exits Example 3. The subset of Ex. 2 with continuous Example 4. The subset of Ex. 3 with Example 5. The set of continuous f : RR that satisfy

Bases Assume that S is a subset of a vector space V over F Definition: The linear span of S is the set of all linear combinations, with coefficients in F, of elements in S Definition: S is linearly independent if Definition: S is a basis for V if S is linearly independent and <S> = V

Problem Set 2 Show that the columns of the d x d identity matrix over F is a basis (the standard basis) of 2. Show examples 1-5 are vector spaces over R 3. Which examples are subsets of other examples 4. Determine a basis for example 1 5. Prove that any two basis for V either are both infinite or contain the same (finite) number of elements. This number is called the dimension of V

Linear Transformations Definition: If V and W and vector spaces over F a function L : V W is a linear transformation if Definition: for positive integers m and n define For every define by (matrix-vector product)

Problem Set 3 Assume that V is a vector space over F and is a basis for V. Then use B to construct a linear transformation from V to that is 1-to-1 2. If V and W are finite dimensional vector spaces over a field F with bases and L : V  W is a linear transformation, use the construction in the exercise above and the definitions in the preceding page to construct an m x n matrix over F that represents L

Discrete Fourier Transform Matrices Definition: for positive integers d define where

Translation and Convolution Definitions: If X is a set and F is a field, F(X) denotes the vector space of F-valued functions on X under pointwise operations. If X is a group and we define translations If X is a finite group we define convolution on F(X) Remark: in abelian groups we usually write gx as g+x and as -g

Problem Set 4 Show that is isomorphic to and translation by is represented by the matrix 2. Show that the columns of the matrix are eigenvectors of multiplication by C 3. Compute 4. Show that where 5. Derive a relation between convolution and

Problem Set 5 Show that the subset of defined by polynomials of degree is a vector space over C. 2. Compute the dimension of by showing that its subset of functions defined by monomials is a basis. 3. Compute the matrix representation for the linear transformation where 4. Compute the matrix representation for translations 5. Compute the matrix representation for convolution by an integrable function f that has compact support. Hint: the matrix entries depend on the moments of f .