1. MA5242 Wavelets Lecture 1 Numbers and Vector Spaces
Wayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore
Tel (65)

2. 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 )

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

4. 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:

5. 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:

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

7. 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

8. 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

9. 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

10. 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)

11. 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

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

13. 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

14. 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

15. 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 .