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%)

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

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)