Spring 2010
  • MATH 304-503: Linear Algebra
  • Time and venue:   TR 12:45-2:00 p.m., MILN 317

    First day hand-out

    Office hours:
  • Tuesday, 11:00 a.m.-12:30 p.m.
  • Thursday, 11:00 a.m.-12:30 p.m.
  • by appointment

  • Help sessions:
  • Mondays - Thursdays, 7:00-9:30 p.m., ENPH 215

  • Office hours before the final exam:
  • Tuesday, May 4, 11:00 a.m.–1:00 p.m.
  • Thursday, May 6, 11:00 a.m.–1:00 p.m.
  • Tuesday, May 11, 2:00–4:00 p.m.

  • Final exam:  Wednesday, May 12, 8:00–10:00 a.m., MILN 317

    Rules for the exam:  no books, no lecture notes, no calculators.  Bring paper and a stapler.

    Sample problems for the final exam (Solutions)

    Sample problems for Test 1 (Solutions)

    Sample problems for Test 2 (Solutions)


    Homework assignments ##1-12


    Course outline:

    Part I: Elementary linear algebra

  • Systems of linear equations
  • Gaussian elimination, Gauss-Jordan reduction
  • Matrices, matrix algebra
  • Determinants

    Leon's book: Chapters 1-2
    Lecture 1: Systems of linear equations.
  • Leon 1.1

  • Lecture 2: Gaussian elimination. Row echelon form. Gauss-Jordan reduction.
  • Leon 1.1-1.2

  • Lecture 3: Some applications of systems of linear equations. Matrix algebra.
  • Leon 1.2-1.3

  • Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.
  • Leon 1.3

  • Lecture 5: Inverse matrix (continued).
  • Leon 1.3-1.4

  • Lecture 6: Transpose of a matrix. Determinants.
  • Leon 1.3, 2.1-2.2

  • Lecture 7: Evaluation of determinants. Cramer's rule.
  • Leon 2.1-2.3


  • Part II: Abstract linear algebra

  • Vector spaces
  • Linear independence
  • Basis and dimension
  • Coordinates, change of basis
  • Linear transformations

    Leon's book: Chapters 3-4
    Lecture 8: Vector spaces. Subspaces.
  • Leon 3.1-3.2

  • Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
  • Leon 3.2

  • Lecture 10: Linear independence. Basis of a vector space.
  • Leon 3.3-3.4

  • Lecture 11: Basis and dimension.
  • Leon 3.4

  • Lecture 12: Rank and nullity of a matrix.
  • Leon 3.2, 3.6

  • Lecture 13: Review for Test 1.
  • Leon 1.1-1.4, 2.1-2.2, 3.1-3.4, 3.6

  • Lecture 14: Basis and coordinates. Change of basis. Linear transformations.
  • Leon 3.5, 4.1

  • Lecture 15: Kernel and range. General linear equation. Matrix transformations.
  • Leon 4.1-4.2

  • Lecture 16a: Matrix of a linear transformation. Similar matrices.
  • Leon 4.2-4.3


  • Part III: Advanced linear algebra

  • Orthogonality
  • Inner products and norms
  • The Gram-Schmidt orthogonalization process
  • Eigenvalues and eigenvectors
  • Diagonalization

    Leon's book: Chapters 5-6 (selected sections)
    Lecture 16b: Euclidean structure in Rn.
  • Leon 5.1

  • Lecture 17: Euclidean structure in Rn (continued). Orthogonal complement. Orthogonal projection.
  • Leon 5.1-5.2

  • Lecture 18: Orthogonal projection (continued). Least squares problems. Normed vetor spaces.
  • Leon 5.2-5.4

  • Lecture 19: Inner product spaces. Orthogonal sets. The Gram-Schmidt process.
  • Leon 5.4-5.6

  • Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors.
  • Leon 5.5-5.6, 6.1

  • Lecture 21: Eigenvalues and eigenvectors (continued). Characteristic polynomial.
  • Leon 6.1, 6.3

  • Lecture 22: Diagonalization. Review for Test 2.
  • Leon 3.5, 4.1-4.3, 5.1-5.6, 6.1, 6.3


  • Part IV: Topics in applied linear algebra

  • Matrix exponentials
  • Rotations in space
  • Orthogonal polynomials
  • Fourier series
  • Markov chains

    Leon's book: Chapters 5-6 (selected sections)
    Lecture 23: Matrix exponentials.
  • Leon 6.2-6.3

  • Lecture 24: Complexification. Orthogonal matrices. Rotations in space.
  • Leon 5.5, 6.1, 6.3-6.4

  • Lecture 25: Orthogonal polynomials.
  • Leon 5.7

  • Lecture 26: Review for the final exam.
  • Leon 1.1-1.4, 2.1-2.2, 3.1-3.6, 4.1-4.3, 5.1-5.7, 6.1-6.3