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.