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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 [email protected] Tel (65) 6874-2749 Z Q R Numbers integers, is a ring rationals, Z p integers modulo a prime, are fields reals, is a complete field under the topology induced by the absolute value | x | x , x R 2 C complex numbers, is a complete field under the topology induced by the absolute value | x iy | x y , x, y R 2 2 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 i C x iy re , r x y 2 i e cos i sin Euler’s Formula y 2 r x Cartesian Geometry Problem Set 1 1. State the definition of group, ring, field. 2. Give addition & multiplication tables for Z2 , Z3 , Z4 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: | uw | | u | | w |, u, w C | u w | | u | | w |, u, w C Vector Spaces over a Field Definition: V is a vector space over a field F if V is an abelian (commutative) group under addition a F , a : V V a ( x y) a ( x) a ( y), x, y V , and, for every this means that a is a homomorphism of V into V, a, b F , u V a b a b a b ab 1u u this means that a a is a ring homomorphism Convention: au a u, a C , u V and, for every d Examples of Vector Spaces a positive integer, F a field v1 v 2 d F v : v j F , j 1,..., d vd with operations d (u v) j u j v j , u, v F (au ) j au j , a F is a vector space over the field F Examples of Vector Spaces Example 1. The set of functions f : R R below f : a, R f ( x) a sin(x ), x R f exits Example 3. The subset of Ex. 2 with continuous f Example 2. The set of f : R R such that Example 4. The subset of Ex. 3 with f ( x) f ( x), xR Example 5. The set of continuous f : RR that satisfy f ( x 2 ) f ( x), xR 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 S n c x : x S , c F j j j j j 1 Definition: S is linearly independent if T S T S S T Definition: S is a basis for V if S is linearly independent and <S> = V Problem Set 2 1. Show that the columns of the d x d identity matrix over F is a basis (the standard basis) of F d 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 L( x y ) Lx Ly, x, y V L(ax) aLx, a F , x V Definition: for positive integers m and n define m n matrices over F n m mn For every A F define LA : F F n by LAu Au, u F (matrix-vector product) F mn Problem Set 3 1. Assume that V is a vector space over F and BV u1 ,, un is a basis for V. Then use B to construct a linear n transformation from V to F that is 1-to-1 2. If V and W are finite dimensional vector spaces over a field F with bases BV u1 ,, un BW w1 ,, wm 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 d 1 1 1 1 2 2 4 1 1 d 1 d 1 2 d 2 2 i / d where e 1 d 1 2d 2 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 g : F ( X ) F ( X ), g X ( g f )( x) f ( g x), f F ( X ), x X 1 If X is a finite group we define convolution on F(X) f h ( x) gX 1 f ( g )h( gx), f , h F ( X ), x X Remark: in abelian groups we usually write gx as g+x 1 and g as -g Problem Set 4 d F ( Z d ) is isomorphic to F and translation by 1 Z d is represented by the matrix 0 0 0 1 2. Show that the columns of the matrix 1 0 0 0 are eigenvectors d C 0 1 0 0 of multiplication by C 2 3. Compute d 0 0 1 0 T 4. Show that I where d d d d d 5. Derive a relation between convolution and d 1. Show that Problem Set 5 1. Show that the subset Pn of C (R ) defined by polynomials of degree n is a vector space over C. 2. Compute the dimension of Pn by showing that its subset of functions defined by monomials is a basis. 3. Compute the matrix representation for the linear d transformation D : Pn Pn where D dx 4. Compute the matrix representation for translations r : Pn Pn , ( r f )( x) f ( x r ) 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 .