Our final exam will be conducted from 10:30am to 12:30pm on Saturday, December 16, 2006, at the usual classroom, i.e., HART 1150.

The final exam is cumulative, i.e., its coverage is the entire course including the materials covered in the midterm exam. In other words, the coverage is:

• All the coverage for the midterm exam; and
• Sections 5.5, 5.6 (from page 320 to 324), 5.9, 5.11, 5.12, Lecture Notes 1 and 2, and 5.13.

Finally, I would like to point out the particularly important subjects we covered in class after the midterm exam.

• Difference between Classical Gram-Schmidt and Modified Gram-Schmidt procedures
• QR factorizations using Gram-Schmidt procedure
• Definitions and properties of unitary and orthogonal matrices as well as hermitian and symmetric matrices
• Complementary subspaces:
  • its definition and concept
  • projections and projectors, properties of projectors
  • complementary projectors
• Orthogonal decomposition of an inner product space:
  • orthogonal complement of a subspace
  • orthogonal decomposition theorems
  • URV factorization
• Singular Value Decomposition (SVD):
  • its definition and its concept (a sphere is mapped to a hyperellipse)
  • computation of SVD for a given matrix
  • computation of matrix norms via singular values
  • low rank approximation and its applications, in particular, image approximation and search engines
  • the pseudoinverse of a given matrix, the Moore-Penrose condition
• Orthogonal projection:
  • difference between an oblique projection and an orthogonal projection
  • construction of the orthogonal projector onto the four fundamental subspaces of a given matrix using that matrix and its pseudoinverse
  • the least squares solution via orthogonal projection

Naoki Saito