Bootcamp in Linear Algebra, Summer 2015
Time and place: MTWTh, 2pm-4pm, MS 5138.
Instructor: Mario Bonk
Office: MS 6137
Office hours: MTWTh after lectures and by appointment
E-mail: mbonk at math.ucla.edu
Phone: (310) 825-4948
Discussion session: Fr, 2pm-4pm, with Jacob Rooney
Textbook: P. Petersen, Linear Algebra, Springer, New York, 2012.
Course material: The bootcamp is intended to help incoming graduate students
with passing the Basic Exam
and prepare them for their future studies at UCLA. This part of the bootcamp covers Linear Algebra. The topics discussed includes the list of subjects relevant
of the Linear Algebra part of the Basic Exam as specified on the
Copies of Basic Exams from previous years can be found here:
Past Qualifying Exams.
Material covered: (this is updated as the course progresses)
Week 1: Basic set-theoretic concepts, groups, rings, fields, vector spaces,
subspaces, linear independence, span of a set, bases, dimension, Zorn's lemma,
dimension formula for sums and intersections of subspaces, linear maps, kernel and image of a linear map, rank and nullity, Rank-Nullity Theorem, vector space isomorphisms, characterization of finite-dimensional vector spaces up to isomorphism.
Week 2: Quotient spaces, spaces of linear maps, dual spaces, matrices,
operations with matrices, the linear map associated with a matrix, coordinates, matrix representations of a linear map, invertible matrices, linear equations, row operations, the Gauss-Jordan elimination algorithm, characterizations of invertible matrices, computation of inverse matrices, coordinate transformations and change of basis, dual maps, dual bases, dual maps and the transpose of a matrix.
Week 3: Traces, permutations and the symmetric group, sign of a permutation, Leibniz formula for determinants, multilinearity, characterization of the determinant function, basic properties of the determinant, Laplace's cofactor expansion, the classical adjoint and inverse matrices, Cramer's rule, determinants and volume, derivatives of matrix-related functions, alternating forms, volume forms, wedge product,
Grassmann algebra, pull-back of forms, differential forms, exterior differentiation, Stokes' theorem.
Week 4: Inner product spaces, norms, Cauchy-Schwarz inequality, angles, orthogonal and orthonormal sets, orthonormal bases,
Gram-Schmidt orthonormalization process, structure theorem for finite-dimensional inner product spaces, metric and topology on an inner product space, Hilbert spaces, completion of inner product spaces, orthogonal projections, orthogonal complements, Riesz representation theorem, adjoints, characterization of orthogonal projection, normal form problem, eigenvectors and eigenvalues, diagonalizable linear maps and matrices,
characteristic polynomial, algebraic and geometric multiplicity of eigenvalues, criteria for diagonalizability, minimal polynomial, invariant and cyclic subspaces, Cayley-Hamilton theorem, normal and self-adjoint
Week 5: Unitary and orthogonal linear operators and matrices,
diagonalizability of symmetric matrices, quadratic forms, criteria for positive definiteness of quadratic forms, Jordan canonical form, nilpotent linear operators.
Homework assignments: (this is optional, but I strongly recommended that you work on these problems; the sections and problems refer to Petersen's book)
Homework set 1: Sect. 1.4, Prob. 5, 6, 8; Sect. 1.5, Prob. 3, 4, 10;
Sect. 1.6, Prob. 1, 2, 5, 6; Sect. 1.8, Prob. 1, 2, 6, 10; Sect. 1.9, Prob. 1, 5.
Homework set 2: Sect. 1.7, Prob. 4, 6; Sect. 1.9, Prob. 4, 5, 6;
Sect. 1.10, Prob. 6, 7, 8; Sect. 1.11, Prob. 2, 3, 9, 14; Sect. 1.12, Prob. 6;
Sect. 1.13, Prob. 9.
Homework set 3: Sect. 1.14, Prob. 4, 6, 11; Sect. 1.15, Prob. 2, 3;
Sect. 5.2, Prob. 1; Sect. 5.3, Prob. 1, 2; Sect. 5.4, Prob. 2, 3, 4, 6, 8.
Homework set 4: Sect. 3.2, Prob. 3, 10; Sect. 3.3, Prob. 7, 9, 10;
Sect. 3.4, Prob. 1, 5, 7; Sect. 3.5, Prob. 8, 10; Sect. 5.5, Prob. 1, 7, 8; Sect. 5.6, Prob. 1, 5; Sect. 5.7, Prob. 5, 10.