Introductory Exam Syllabus

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).

Combinatorial analysis
Inclusion-exclusion
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).

Basic set theory:
Finiteness, countability, and uncountability
Ordered fields
Cauchy sequences
Construction of the real numbers
Completeness of the real numbers
Infimum and supremum
Liminf and limsup of sequences
Limit points of sets, and sequences
Norms, inner products, and metrics
Open sets and closed sets
Closure and boundary of a set
Compact sets
Bolzano–Weierstrass theorem
Heine–Borel theorem in Euclidean space
Intersection of nested compact sets

Continuous mapping:
Sums, products, compositions of continuous mappings
Path-connected and connected sets
Images of compact and connected sets
Extreme value theorem
Intermediate value theorem
Uniform continuity

Differentiability:
Chain rule
Product rule
Mean value theorem
Taylor’s theorem
Derivative matrix
Continuity of differentiable mappings
Differentiable paths
Directional derivatives
Inverse function theorem
Implicit function theorem
Maxima and minima
Constrained extrema and Lagrange multipliers

Integration:
Existence and elementary properties of the Riemann integral
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

Sequences and series:
Numerical sequence and series
Pointwise and uniform convergence of sequence of functions
The Weierstrass M-test
Differentiation and integration of series
Elementary functions (exponential, trigonometric, hyperbolic)
Contraction mapping theorem
The Stone-Weierstrass theorem
Dirichlet and Abel tests
Power series and Cesàro and Abel summability

REFERENCES

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)
The Way of Analysis, by R. S. Strichartz. (Jones & Bartlett, 1995)