Linear Algebra I
G63.2110
Fall 2005, Mondays, 5:10 pm - 7:00 pm.
Instructor:
Olof Widlund (101, 19 University Place)
Coordinates of Olof Widlund
Office: WWH 712
Telephone: 998-3110
Office Hours: Mondays 4:00 - 5:00 pm. You can also try to drop in
or send email or call for an appointment.
Email: widlund@cs.nyu.edu
Text book: Linear Algebra by Friedberg, Insel, and Spence. Prentice
Hall. Homework assignments will often be from the Fourth Edition of this
book.
Homework
There will be regular homework assignments.
It is important that you do the homework yourself, but when you get stuck,
I encourage you to consult with other students
or me, to get help when necessary. However, when you get help,
it's important to acknowledge it in writing.
Please staple everything together and order the problems in the
same order as given. The best way of turning in homework is to give it to me
personally, in class or in my office. If I am not in my office, you can
slide it under my door. If you put your homework in my mailbox, it is at
your own risk.
The grader Jose Cal Neto is now available in WWH 711 to answer questions
concerning the homework. He will be there Mondays 4:00-5:00pm and
Wednesdays 6:00-7:00pm.
October 10 is Columbus Day, which is a university holiday.
There will be a class on Wednesday, November 23 to make up
for Columbus Day. It is a so called Legislative Day. All
classes that day runs on a Monday schedule.
The last class
will be held on December 12.
Final examination will take place on Monday, December 19,
5:00-6:50pm in the same class room where we meet regularly.
You may bring two sheets of papers (regular size) covered front and
back by notes for your exclusive, private use during the examination.
(The course content is defined by the notes on the lectures given below.)
Homework Assignments:
Home work set 1, due October 3: 3 p. 6; 12 p. 15; 8 p. 20; 19 p. 21;
3 p. 33; 5 p. 41; 14 p. 42; 13 p. 55; 24 p. 56; and 28 p. 57;
all from the text book.
Home work set 2, due October 3, but without penalty it can be turned in
on October 17: 4 p. 74; 9, 10 p. 75; 20, 21 p. 76; 25 p. 77; 35 p. 78.
Home work set 3, due October 17: 3 p. 84; 6 p. 85; 14, 16, p. 86;
8, 13 p. 97; 15, 17 p. 98.
Home work set 4, due October 31: 6, 7, 9, 11, and 13 p. 107; 17 p. 108;
from the handout: 20.2, 20.3, 21.2, 21.5, and 21.6.
Home work set 5, due November 7: 2, 6, 8, and 12 on pp. 165-169;
2, 3, and 7 on pp. 179-181; 4, 8, and 10 on pp. 196-197.
Home work set 6, due November 14: 3, 9, 12, 20, 22, 24, and 28
on pp. 228-232.
Home work set 6 b, due November 14: 11 on p. 208; 1 on p. 236.
Home work set 7, due November 28: 4, 11, 18 on pp. 141-144; 3, 8, 9,
12, 14, 20 on pp. 257-260; 2, 8, 11, 18 on pp. 279-282.
Home work set 8, due December 5: Problems 3.3 and 3.5 on last page of the
handout on norms given out on November 23; 5, 8, 10 on pp. 309-310; 3, 8, 11, 17 on
pp. 336-338; 4, 8, 11 on p. 354.
Home work set 9, due December 12: Problems 3, 7, 20, and 22
on pp. 366-368, 2, 4, 13, and 17 on pp. 375-377, and 2, 5, 21, and 24 on
pp. 392-395.
Lectures
September 12: Vector spaces. Examples of vector spaces which satisfies
the properties required. Subspaces and how to verify that we have a
subspace. Linear combinations and the span of a finite set.
September 19: Various results from sections 1.4 - 1.6 of the text book:
Determining if a element is in the linear span of a set of elements by
solving a linear system of algebraic equations. Generating sets and bases
of linear spaces. The replacement theorem and corollaries. Bases and dimension
of linear spaces.
September 26: Further discussion of the consequences of the replacement theorem.
Polynomial interpolation; the Lagrange and the cubic Hermite case. Linear transformations
and examples of such transformations. The dimension theorem.
October 3: Functions and functions that are one-to-one, onto, and
invertible. Necessary and sufficient conditions that a linear transformation
is one-to-one. Linear mappings between linear spaces of the same dimension.
Defining linear transformations in terms of the images of the elements in
a basis of a linear space. Matrix representation of linear transformations;
ordered basis of linear spaces. The linear space of linear transformations
between a pair of linear spaces. Products of linear transformations and
of matrices.
October 10: Columbus Day NYU holiday.
October 17: (There was a handout given out. If you need it and could not
attend class, let me know and I will mail it to you. You can also pick it up
directly from me in my office.)
Inverses of linear transformations and matrices.
Computing inverses of square matrices by solving linear systems of algebraic
equations. Gaussian elimination: the case without pivoting and partial pivoting.
The relation between Gaussian elimination and factoring matrices into triangular
factors. Solving linear systems with triangular coefficient matrices. The cost
of Gaussian elimination.
October 24: Solving linear systems of algebraic equations, continued;
essentially the entire chapter 3. Linear transformations that preserve
rank. Isomorphisms and the isomorphism of
families of linear transformations and matrices.
October 31: (There was a handout on Cholesky' method for positive
definite, Hermitian matrices.) Hermitian and symmetric matrices. Positive
definite Hermitian matrices. Cholesky's algorithm. Determinants. The two-by-two
case. Relations with area of parallelograms. The recursive definition of
determinants for arbitrary square matrices and some first results.
November 7: (There is a handout with solutions to most of the problem
set 4.) Determinants of n-by-n matrices as in section 4.2 of the text.
Finding the null space of differential operators of order n with constant
coefficients; to be continued.
November 14: (A handout with the solutions of some of problem set 5
was given out.) Finding the null space of differential operators of order n
with constant coefficients; conclusion. Eigenvalues and eigenvectors of
matrices. Transforming a matrix to diagonal form by using a complete set
of eigenvectors. Linear independence of eigenvectors and the existence of
a full set of eigenvectors if all eigenvalues are distinct. Example of
matrices which have fewer eigenvectors and which cannot be transformed to
diagonal form. A proof that the dimension of the eigenspace of an eigenvalue
is at least 1 and never exceeds the multiplicity of the eigenvalue.
November 21: Computing eigenvectors and eigenvalues using the power
method and inverse power method with shift. Similarity transformations; they
preserve the eigenvalues. Limits of matrix sequences. Stochastic matrices.
The 1- and infinity-norms of vectors and their related matrix norms. Using
matrix norms to estimate the largest eigenvalue of a matrix. Gerschgorin's
disk theorem. A few words on inner products.
November 23: (Handout on norms of vectors and matrices.) Inner product
spaces. Gram-Schmidt. Orthogonal inequalities. Bessel's inequality. Linear
least squares problems.
November 28: (Handout on Householder transformations and of a practice final.)
Linear least square problems; fitting linear and quadratic polynomials.
Householder transformations and the QR factorization of matrices. The use of QR
factorizations to solve least squares problems. Sensitivity of the solution of
linear systems to perturbations of the right hand side; an application of
vector and matrix norms.
December 5: (Handout on singular values.) Representing linear operators
using orthogonal bases. The definition of adjoints. Eigenvalues of
operators and their adjoints. Schur's normal form of square matrices. Normal
linear operators and matrices. Eigenvector systems of normal matrices. Hermitian
and skew-Hermitian matrices. Introduction to the singular value decomposition.
December 12. Discussion of the practice final. Remarks on the singular
value decomposition; determination of rank, solving least squares problems.