Syllabus for the Introductory Exam
Problems in the exam will be drawn from the following three areas in roughly the following proportions
- Linear Algebra (33 1/3%)
- Probability (33 1/3%)
- Real Analysis (33 1/3%)
LINEAR ALGEBRA
Vectors & Analytic Geometry of Space
Coordinate Systems in n-space
n-Vectors
Linear
Dependence of n-vectors
Length & Inner
Product of n-vectors
Outer/Cross
Product of 3-vectors
Finite-Dimensional Vector Spaces
n-dimensional
Vectors Spaces Over R and C
Linear
Dependence & Generators
Simultaneous
Linear Equation, Gaussian elimination
Bases & Dimensions
Subspaces
Inner
Products; Schwarz Inequality
Orthogonal
Bases; Orthogonal Complements; Projections
Dual
Spaces
Lines,
Hyperplanes, Coordinate systems
Distance;
Polarization Identity
Linear Transformations and Matrices
Linear
Transformations; Elementary Properties (Image/Nullspace)
Addition,
Composition, Scalar Multiplication
Matrices
\& Linear Transformations
Nonsingular
Linear Transformations; Inverses
Changes
of Bases in Vector Spaces
Similarity
of Matrices
Special Types of Square Matrices (Symmetric, Orthogonal, Triangular etc.)
Elementary Matrices
Rank
of a Matrix
Determinants
Definition & Elementary
Properties
Permutations & Uniqueness
Minors;
Cofactors; Evaluation of Determinants
Applications
(Dependence of Vectors; Volume of a Parallelopiped; Linear Equations)
Determinants
of Products & Inverses
Determinants of Special Types of Matrices
Bilinear and Quadratic Forms
Bilinear Mappings & Forms
Quadratic
Forms; Polarization
Equivalence
of Quadratic Forms; Congruence of Matrices
Geometric
Applications
Eigenvalues and Eigenvectors
Definitions
Similarity & Diagonal
Matrices
Orthogonal
Reduction of Symmetric Matrices
Eigenvalues
of Special Types of Matrices (Unitary, Hermitian, etc.)
Normal
Matrices & Spectral Theorem
Singular
Values
References
Elements of Linear Algebra, by L. J. Paige and J. D. Swift
(Ginn & Co, 1961)
Finite-Dimensional Vector Spaces, by P. R. Halmos (D. Van Nostrand,
1958)
Introduction to Linear Algebra, by S. Lang (Springer-Verlag
UTM 1986)
Linear Algebra, by K. Jänich (Springer-Verlag UTM 1994).
PROBABILITY
Combinatorial analysis
Permutations
Combinations
Multinomial coefficients
Axioms of probability
Sample spaces and events
Axioms of probability
The uniform model
Probability as a continuous set function
Conditional probability and independence
Conditional probabilities
Bayes' theorem
Independent events
Random variables and their distributions
Random variables
Discrete random variables, Indicator functions
Expected value
Expectation of a function of a random variable
Variance
Distribution function
Probability mass function
Probability density function
Discrete univariate distributions
Bernoulli and binomial distributions
Geometric and negative binomial distributions
Poisson distributions
Hypergeometric distributions
Continuous univariate distributions
Uniform distributions
Normal distributions
Exponential distributions
Gamma distributions
Cauchy distributions
Change of variables
Multivariate distributions
Independent random variables
Convolution and sums of independent random variables
Conditional distributions
Change of variables
Multivariate normal distributions
Properties of expectation and related matters
Linearity of expectation
Covariance
Variance of a sum
Correlation
Conditional expectation and the law of total expectation
Conditional variance and the law of total variance
Conditional covariance and the law of total covariance
Conditional expectation and prediction
Moment generating functions
Inequalities and limit theorems
Chebychev's
and Markov's inequalities
Cauchy-Schwarz inequality
Jensen's inequality
Weak and strong laws of large numbers
Central limit theorem
References
A First Course in Probability, by S. Ross (Prentice Hall, 2001).
Essentials of Probability, by R. Durrett (Duxbury, 1993).
REAL ANALYSIS
Basic Set Theory
Sets & Functions
Boolean operations (unions, intersections, complements) and basic properties
De Morgan's laws
Product sets
Finite, countable and uncountable sets
The Real Line
Ordered Fields
Completeness & The Real Number System
Least Upper Bounds
Cauchy Sequences; Cluster Points; liminf and limsup
Euclidean Space
Norms; Inner Products; Metrics
Complex Numbers
The Topology of Euclidean Space
Open & Closed Sets
Accumulation
Points; Interior, Closure and Boundary of a Set
Sequences & Series
in Euclidean Space; Completeness
Compactness;
Heine-Borel Theorem; Intersection of Nested Compact Sets
Path-Connected & Connected
Sets
Continuous Mappings
Continuity;
Images of Compact \& Connected Sets
Operations
on Continuous Mappings (Sums, Products, Compositions)
Boundedness
on Compact Sets
Intermediate
Value Theorem
Uniform
Continuity
Elementary
Differentiation & Integration
Uniform Convergence
Pointwise & Uniform Convergence
The
Weierstrass M-Test
Differentiation & Integration
of Series
Elementary
Functions (Exponential, Trigonometric, Hyperbolic)
The
Space of Continuous Functions
Equicontinuity & the
Arzela-Ascoli Theorem
Contraction Mapping Theorem
Stone-Weierstrass
Theorem
Dirichlet & Abel
Tests
Power
Series
Cesaro & Abel
Summability
Differentiable Mappings
Derivative Matrix; Continuity of Differentiable Mappings
Differentiable
Paths; Directional Derivatives
Chain
Rule; Product Rule
The
Mean-Value Theorem
Higher
Derivatives & Taylor's Theorem
Maxima & Minima
Inverse & Implicit Function Theorems
Inverse Function Theorem
Implicit
Function Theorem
Rectifiability
of Curves and Surfaces; Existence Theorem for ODE's
Morse
Lemma
Constrained Extrema & Lagrange Multipliers
Integration Theory
Existence of the Riemann-Integral; Elementary Properties
Fundamental
Theorems of Calculus
Differentiation of Limits of Integration
Interchange
of Order of Integration (Fubini's theorem)
Differentiation
of Integrals with respect to a Parameter in the Integrand
Change
of Variables Theorem; Polar, Spherical and Cylindrical Coordinates
Improper
Integrals
Elementary
Numerical Quadrature (Midpoint, Trapezoidal, Simpson Rules)
Fourier Series
Inner Product Spaces; Orthogonal Families of Functions; Completeness
Calculation
of Fourier Coefficients
Convergence Theorems (Uniform for C1, Mean for L2)
Differential equations
Separable equations
Linear, homogeneous, constant coefficient equations
References
The Way of Analysis, by R. S. Strichartz. (Jones & Bartlett,
1995)
Elementary Classical Analysis, 2nd ed., by J. E. Marsden & M.
J. Hoffman (W. H. Freeman, 1993)
Principles of Mathematical Analysis, 3rd ed., W. Rudin (McGraw
Hill, 1976)