Linear Algebra I

Fall 2011, Tuesdays, 5:10 - 7:00 pm, Warren Weaver Hall 517

Instructor: Olof Widlund

  • Coordinates of Olof Widlund
    Office: WWH 612
    Telephone: 998-3110
    Office Hours: Tuesdays and Thursdays, 4:00 - 5:00 pm. You can also try to drop in or send email or call for an appointment.
    Email: widlund at
    URL for this course:

  • 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. Scores will be available on Blackboard. 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 is 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.

  • Homework Assignments:
  • Set 1, due September 27 at midnight: #12 and 18 on p. 15; #10 and 12 on p. 21; #28 and 30 on p. 23; #12 on p. 34; #15 on p. 35; # 10 on p. 41; #17 on p. 42; #11 on p. 55; #16 on p. 56; #29 and 31 on p. 57.
  • Set 2, due October 4 at midnight: #9 and 11 on p. 75; #20 on p. 76; #26 on 77; #35 on p. 78; #7 on p. 85 and #15 on p. 86.
  • Set 3, due October 18 at 5:10pm: #9 and 13 on p. 97; #17 on p. 98; #6 and 9 on p. 107; #16 on p. 108 and #22 on p. 109.
  • Set 4, due October 25 at midnight: #20.2, 20.3, 21.2, 21.5, and 21.6, all from the handout given out on October 4.
  • Set 5, due November 9 at 10:00am: #4, 6, and 8 on pp. 124-125; #3b and 3d on p. 141; #2b, 2d, 2f, 6b, and 6d on pp. 165-167; #3b, 3d, 5, and 8 on pp.180-181.
  • Set 6, due November 16 at 10:00am: #4 on p. 221; #22 and 25 on p. 222; #4 on p. 228; #12 and 15 on p. 229; #23 on p. 230; #9 on p. 258; #14 and 18 on p. 259.
  • Set 7, due November 23 at 10:00am: #2b and f, 8, 14a, 18 on pp. 279-282; 21 and 23 on p. 312.
  • Set 8, due December 7 at 10:00am: #3, 5, 17, and 19 on pp. 322-324; #8, 11, and 24 on pp. 337-339.

  • Blackboard: That system will be used primarily for the homework and exam scores. A grader has been appointed on September 19.

  • Exams: There will be a midterm exam on October 18 and a final exam on December 20. You may bring two standard sheets of paper to the exams with notes written by yourself on both sides and for your own private use during the exam.

  • Lectures
  • September 6: 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. Linear dependence and linear independence. The span of a set of elements is a linear space V; it is a subspace of V.
  • September 13: Bases for spaces of polynomials. Lagrange and Hermite interpolation. These procedures provide alternative bases, which are convenient when solving interpolation problems. Sums and direct sums of sets and vector spaces. Generating sets and bases for vector spaces; a basis provides a minimal generating set and all elements of the generated space are represented uniquely as a linear combination of the basis elements. Finite dimensional vector spaces; any two bases of such a space have the same number of elements. The replacement theorem and a variety of applications thereof. Examples of linear spaces of matrices and explicit bases for these spaces which allow us to compute their dimensions. (This brings us to the end of our discussion of Chapter 1 of the textbook.)
  • September 20: Linear transformations; functions on vector spaces preserving the linear structure. Example: differentiation, integration, rotations, reflections, and projections. Range and null spaces of linear transformations; nullity and rank. nullity(T)+rank(T)=dim(V), for V of finite dimension. one-to-one and onto functions and linear transformations. Matrix representation of linear transformations and ordered bases. The set of all linear transformations between a pair of vector spaces is a vector space and so are the set of m-by-n matrices. Composition of linear transformations and matrix multiplication.
  • September 27: More on linear transformations. Isomorphisms and invertibility of linear transformations and matrices. The isomorphism of families of linear transformations and matrices. A first introduction to Gaussian elimination; a handout will be available next week.
  • October 4: Handout on Gaussian Elimination. Solving linear systems of algebraic equations. Straight lines, planes, or hyperplanes that intersect in one point or not. Comments on the column and row ranks of matrices; we will establish that they are the same. Gaussian elimination in terms of special unit lower triangular matrices. The inverses of these matrices and products of these matrices can easily be found. The need for pivoting and the partial pivoting strategy; if the matrix in invertible, this strategy will always work. What happens if the matrix is not invertible. Cost of Gaussian factorization in the general case and in the case when the matrix is tridiagonal. The cost of solving the system of equations once the triangular factors are known.
  • October 11: NYU holiday.
  • October 18: Midterm exam. The exam, with solutions.
  • October 25: Material from Chapter 3, in particular a proof and a discussion of theorems 3.4 and 3.6. Homogeneous linear differential equations with constant coefficients. Note that there is a complication since the relevant vector space is not finite dimensional; see section 2.7. A few words on linear functionals; see section 2.6.
  • November 1: More about dual spaces and linear functionals. The second dual of a finite dimensional linear space V is isomorphic to V. Determinants of 2-by-2 matrices and the recursive definition of determinants in terms of determinants of smaller matrices. Cofactors and cofactor expansions with respect to an arbitrary row of the matrix; this requires proofs by induction.
  • November 8: Assorted results on determinants: the effect of interchanging rows, the determinant of a product of two matrices, the determinant of the transpose of a matrix, among them. Determinants and the area or volume of parallelograms and parallelepipeds. Eigenvalues and eigenvectors of matrices. Transforming matrices onto diagonal form by using their eigenvectors; it does not always work because a full set of eigenvectors is not available for all matrices.
  • November 15: Alternatives for matrices that cannot be transformed into diagonal form: a brief discussion of the Jordan normal form and the Schur factorization of matrices. Limits of sequences of matrices and sequences obtained by powers of a given matrix. Conditions on the eigenvalues of matrix in order for its powers to converge to a limit. Transition matrices. Vector and matrix norms associated with vector norms. How to compute the ell_1 and ell_infinity norms of matrices. Gerschgorin's circle theorem. Invariant subspaces and invariant subspaces defined by an element in the vector space. The eigenvalues of the linear operator defined by restricting the given operator to such a subspace; these eigenvalues form a subset of the eigenvalues of the given linear operator.
  • November 22: The Caley-Hamilton theorem with a proof for diagonalizable matrices and another for the general case. Inner product spaces with several examples. Orthonormal bases for inner product spaces and the Gram-Schmidt algorithm. Legendre polynomial, Hermite and Laguerre polynomials. Fourier series. Projections onto subspaces spanned by a subset of an orthonormal basis and a few words on linear least squares.
  • November 29: There were two handouts on Householder transformations and symmetric tridiagonal matrices, respectively. Further results on systems of orthogonal polynomials: they can be generated by three-term recursions and all their roots are simple and lie in the interior of the interval of integration. Linear least squares problem and the normal equations derived in two different ways. Factoring matrices in terms of a product of an orthogonal matrix Q and a matrix R with an upper triangular matrix on top. How to compute QR factorizations by Householder transformations and by Gram-Schmidt. How to use Householder transformations to construct an upper Hessenberg matrix which is similar to a given square matrix. The special case of symmetric matrices; the Hessenberg matrix is then symmetric and tridiagonal. How to compute the characteristic polynomial of a symmetric tridiagonal matrix by a three-term recursion.
  • December 6: Cauchy's interlace theorem and Sturm sequences. Schur factorization and normal matrices. A different proof of the fact that two eigenvectors associated with two different eigenvalues of a symmetric matrix are orthogonal. Rayleigh quotient. Formulation of Sylvester's inertia theorem and using it to determine the inertia of a Schur complement of a symmetric matrix. First comments on Courant-Fischer's theorem.
  • December 13: Courant-Fischer's theorem. Sylvester's inertia theorem. the singular value decomposition and a few applications. Solving underdetermined linear systems of equations. A few words in conditioning of linear systems of algebraic equations.