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Paul Skoufranis

Ph.D. Student

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Curriculum Vitae

Summer 2012: Boot Camp

Course Information:

The following is a link to the 'Fall 2012 New Student Information' which contains all pertinent information about a mathematics graduate students life at UCLA up to the commencement of the Fall 2012 quarter.

http://www.math.ucla.edu/grad/F12newstudentinfo.shtml

The following are the links to the dates for the Fall 2012 Mathematics Qualifying Exams and to the database for past Qualifying Exams respectively:
http://www.math.ucla.edu/grad/handbook/exdates-f12.shtml
http://www.math.ucla.edu/grad/handbook/hbquals.shtml

The following is a link to the free textbook used for the linear algebra portion of the boot camp.  Another excellent reference for this material is the fourth edition of Linear Algebra by Friedberg, Insel, and Spence.

http://www.springerlink.com/content/978-1-4614-3611-9/#section=1080576&page=1

Additional Discussion Practice Questions:

The following are some additional questions to work on during discussion.  Some are straightfoward but some are fairly difficult.
Week One Extra Practice Questions
Week Two Extra Practice Questions
Week Three Extra Practice Questions
Week Four Extra Practice Questions
Week Five Extra Practice Questions
Recurring Basic Exam Linear Algebra Problems


Additional Material:

The following are various notes that I have developed on linear algebra topics for the upper division linear algebra course, MATH 115A, at UCLA.  These notes were designed for a student new to mathematical proofs and thus contain all details and attempt to develop ideas as clearly as possible.  Thus these notes may seem to intensely simplify many topics and bore those with advanced mathematical knowledge.  A description of each document is contained next to the link.  Note that some of the documents may not contain all pertinent information about the topic as homework problems were assigned pertaining to that material.

Notes:
Direct Sums of Vector Spaces (Develops the theory of direct sums of vector spaces)
Quotients of Vector Spaces (Develops some theory of quotients of vector spaces)
A More Detailed Proof of Theorem 1.7 from the Text (A proof that if S is a linearly independent set and v is a vector not in S, then S union {v} is linearly independent if and only if v is not in the span of S)
Maximal Linearly Independent Subsets (Draws the connection between bases and maximal linearly independent subsets)
One-to-One and Onto Functions (Develops some basic theory of bijective functions)
Coordinates and the Matrix of a Linear Transformation (Develops the theory of coordinates, change of basis matrices, and matrices of linear transformations)
Summary of Invertibility and Isomorphisms (A summary of the theory of invertible linear transformations)
Dual Spaces (Develops some basic properties of dual spaces)
L_A Explained (Explains why left multiplication by a matrix A is an important linear transformation)
Elementary Row Operations Explained (Explains why elementary row operators have the effect they do)
Review of Determinants (Reviews properties and computations of determinants; no proof are given)
Determinant of the Vandermonde Matrix (A computation of the determinant of the Vandermonde matrix using induction)
Summary Of Diagonalizability (A summary of the theory of Diagonlizability of a linear map)
Limits of Matrices (Uses diagonalizability to compute limits of matrices)
Gram-Schmidt Orthogonalization Process and QR Factorization Notes (An example computation and proof of each computation)
Orthogonal Projections Notes (Develops the theory of orthogonal projections on finite dimensional inner product spaces)
The Cayley-Hamilton Theorem and Minimal Polynomials (Mainly a development of the theory of minimal polynomials once the Cayley-Hamilton Theorem has been proved)
The Spectral Theorem for Normal and Self-Adjoint Operators (A complete proof of the spectral theorem for normal operators and real self-adjoint matrices)



The following are some pratice questions and tests that I have developed on linear algebra topics for the upper division linear algebra course, MATH 115A, at UCLA.  Thus they are probably simpler to what you might find on the Basic Exam.

Practice Questions and Tests:
MATH 33A Review Questions (Some basic/common computation problems)
Practice Quiz One (A practice quiz based on Chapter 1 of Linear Algebra by Friedberg, Insel, and Spence)
Practice Midterm (A practice mideterm based on Chapters 1 and 2 of Linear Algebra by Friedberg, Insel, and Spence)
Practice Quiz Two (A practice quize based on Chapters 2 and 5 of Linear Algebra by Friedberg, Insel, and Spence)
Practice Final Exam (A practice final exam based on Chapters 1, 2, 5, and 6 of Linear Algebra by Friedberg, Insel, and Spence)