MATH2701
Abstract Algebra and Fundamental Analysis
Term 3 - 2022
Algebra
Transformations and Groups
- Group
- Abelian Group
- Basic Properties of Groups
- Order
- Group Generator
- Transformation
- Subgroups of Bijections of Plane
Isometries
- Isometry
- Every Isometry is Injective
- Isometry Group
- Isometries of Real Coordinate Space
- Linear Isometry Preserves Inner Product
- General Linear Group
- Orthogonal Group
- Plane Isometries
- Classifying Plane Isometries
- Reflection
- Rotation
- Glide Reflection
- Translation
Congruence and Symmetries
Similarities
- Similarity Transformation
- Dilation
- Collineation
- Dilatation
- Classifying Plane Similarities
- Affine Transformation
More Groups
Real Projective Space
- Real Projective Space
- Duality Principal
- Desargue's Theorem
- Real Projective Transformation
- Projective Closure
Analysis
Asymptotics
Inequalities
- Triangle Inequality
- Cauchy-Schwarz Inequality
- Concave and Convex Functions
- Jensen's Inequality
- AM-GM Inequality
- Young's Inequality
- Holder's Inequality
- Minkowski's Inequality
- Interchange of Function and Limit
Norms and Convex Bodies
- Norm
- Normed Vector Space
- \(\ell^p\) Sequence Space
- \(L^p\) Space
- Convex Body
- Correspondence Between Norms and Symmetric Convex Bodies
- size of norm and unit ball inclusions
- interchange of function and supremum
Duality
Primes
Real Numbers
- Natural Numbers
- Integers
- Rational Numbers
- Algebraic Structure
- Group
- Monoid
- Ring
- Ordered Field
- Cauchy Sequence
- Bolzano-Weierstrass Theorem
- Real Numbers
- Dedekind Cut
- Completion of a Metric Space