MA5242 Wavelets Lecture 1 Numbers and Vector Spaces

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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_at_nus.edu.sg Tel [phone]
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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 )
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Polar Representation of Complex Numbers
Z integer, R-real, Q-rational, C-complex
Polar Representation of the Field
Eulers Formula
Cartesian Geometry
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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
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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
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Examples of Vector Spaces
a positive integer,
a field
with operations
is a vector space over the field
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Examples of Vector Spaces
Example 1. The set of functions f R ? R below
exits
Example 2. The set of f R ? R such that
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
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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 ltSgt V
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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
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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)
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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
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Discrete Fourier Transform Matrices
Definition for positive integers d define
where
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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
gx
and
as -g
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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
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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 .