math222final - Math 222 Fall 2004 Final Exam Closed books...

Math 222, Fall 2004: Final Exam
• Closed books, no notes, no calculators.
• Upper case letters denote matrices, boldface lower case letter vectors,
and plain lower case letters scalars.
• The transpose of a matrix A is denoted A
t
.
• The identity matrix is denoted I .
• A matrix U is said to be orthogonal if U
t
U = I .
• In questions 2 – 6, please provide brief explanations for the steps in
your calculations.
Question 1: There is no connection between the matrices and the vectors
in the following questions. No motivation is required. (2p each)
(a) Suppose that A is an m × n matrix of rank k . What is dim(Col( A ))?
(b) Suppose that there exists a non-zero vector y such that A y = 0.
Does there exist a vector b such that the equation A x = b has a
unique solution?
(c) Is it true that λ is an eigenvalue of A if and only if det( A - λI ) = 0?
(d) Is it true that all square matrices are diagonalizable?
(e) What conditions must a matrix satisfy for it to have a singular value
decomposition?