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Curriculum Arun Ram

School of Mathematics and Statistics University of Melbourne Parkville VIC 3010 Australia aram@unimelb.edu.au

This page is built on the philosophy that the way to specify a curriculum is to specify what kinds of tasks will be expected, i.e. what the assessment will be. In other words, a curriculum is determined, whether we admit it or not, by specifying what questions can appear on the exam. If the students are not expecting a question on the exam, then the likelihood that they will learn to do that question is very low.

Some of the material available from the links below is based upon work supported by the Australian Research Council ARC grants DP0986774 and DP087995 and the US National Science Foundation under Grant No. 0353038 and earlier awards. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of these agencies.

Group Theory and Linear Algebra

Greatest common divisors, Euclid's algorithm, arithmetic modulo m

• Common divisors of 56 and 72
• Show that a|b and a|c implies a^2|(b^2+3c^2)
• Find gcd(323,377)
• Calculations in Z/10/Z, Z/7Z and Z/20Z
• Powers of 3 mod 19
• A condition for divisibility by 11
• Show that if a=b (mod m) and b=c (mod m) then a=c (mod m)
• Addition, multiplication and inverses in Z/7Z
• The smallest positive number in 6Z+15Z
• Show that if ac=bc (mod m) and gcd(c,m)=d then ad=bd (mod m)
• Show that if p is prime then n^p=n mod p

Fields, RSA cryptography

• Determining whether subsets of R and C are fields
• Show that, in a field, c0=0 and if ab=0 then a=0 or b=0
• Algebraic closure, factoring and square roots in F_7
• Inverses and solving equations in Z/35Z and Z/24Z
• Using Fermat's little theorem for 3^{52} (mod 53)
• Using Euler's theorem for 30^{62} (mod 77)
• RSA encryption and decryption

Bases, linear transformations, eigenvalues, direct sums invariant subspaces

• Is S={(1,3), (3,4), (2,3)} in F_5^2 a basis?
• Are {1, sin^ x, cos^2x} and {1, sin(2x), cos(2x)} linearly independent?
• Show that {1, sqrt(2), sqrt(3)} is linearly independent over Q
• Multiplication by a complex number as an R-linear transformation
• F_5^4 as a direct sum of invariant subspaces , Second part
• Eigenvalues and eigenvectors of a 3-cycle

Minimal polynomials, diagonalization

• Using the minimal polynomial to find an inverse
• Finding minimal polynomials
• The Jordan basis when the minimal polynomial is x^n
• The eigenvalues and eigenvectors of the transpose map and another part Perhaps these need to be patched together
• Diagonalisability of f when f^4=1
• Diagonalising f when f^2=f
• Common eigenvectors when fg=gf
• A linear transformation with no minimal polynomial

Jordan normal form

• Finding Jordan form from minimal and characteristic polynomials
• Finding possible Jordan forms from the characteristic polynomial
• Jordan forms, rank(A-alpha) and dimension of an eigenspace
• Determining similar matrices
• Characterizing Jordan form with subspaces

Properties and examples of groups, subgroups, cyclic groups orders of elements

• Matrix groups and matrix non groups
• Solving equations in groups
• The group of nth roots of unity
• Products of permutations in S_6
• Orders of elements in S_5 and C^x
• Cyclic subgroups of S_3
• If g^2=1 then G is abelian

Direct product, homomorphisms and isomorphisms, cosets

• The order of (1,2) in Z/2Z x Z/8Z
• Homomorphisms and nonhomomorphisms from GL(n) to GL(n)
• Conjugation isomorphisms
• Homomorphisms from Z to Z
• Explaining why groups are nonisomorphic
• Left cosets of ⟨ 3 ⟩ in Z/6Z
• Left cosets of ⟨s,r²⟩ in the dihedral group D_6

Normal subgroups, Lagrange's theorem, quotient groups

• Quotients of ⟨(1,0)⟩ and ⟨(0,2)⟩ in Z/2Z x Z/4Z
• Sizes of subgroups and intersections of subgroups
• Quotient groups of cyclic groups are cyclic
• If [G:H]=2 then H is normal.
• The subgroups T, B and U in GL_2(R)
• Determine all subgroups of the dihedral group D_5

Inner products

• Non inner products
• Length of (1-2i, 2+3i) in ℂ²
• Find an orthonormal basis of ℂ² containing (1+i,1-i)
• The complex parallelogram law for an inner product
• The inner product ⟨ v,w ⟩ = Re(v,w)
• The orthogonal complement of span{(0,1,0,1),(2,0,-3,-1)} in ℝ⁴
• W⊆ (W^⊥)^⊥ and W=(W^⊥)^⊥ when dim(V) is finite

Adjoints, spectral theorem

• Self adjoint, isometric and normal matrices
• Finding a unitary U such that U^*AU is diagonal
• Show that ker (f^*) = (im f)^⊥
• Show that eigenvalues of self adjoint f are real and of isometries have absolute value 1
• Show that every normal matrix has a square root
• Show that there exists a unique decomposition A=B+C with B=B^* and C=-C^*

Group actions, orbit-stabiliser theorem, Sylow theorems

• Right multiplication by g^-1 gives group action
• Stabilizers and orbits of vertices and edge midpoints of a rectangle
• Stabilizers and orbits for the GL(2) action on ℝ²
• If |G|=9 and |X|=16 then an action of G on X has a fixed point
• Conjugacy classes and centralisers of (12) and (123) in S₃
• 2, 3 and 7-Sylow subgroups in G when |G|=84
• Show that there is a unique p-Sylow subgroup when G is abelian
• Counting p-Sylow subgroups when |G|=30

Derivatives

Metric and Hilbert spaces

• Baby Rudin Questions Chapter 2
• Baby Rudin Questions Chapter 3
• Baby Rudin Questions Chapter 4
• Bressan Problems Chapter 11
• Some MAST30026 Assignment sheets
• MAST30026 Problem sheets by Hyam Rubinstein