Description
Read textbook pages 135 to 142, pages 126 to 128 before working on the homework problems. Show all steps to get full credits.
1. Let A =
Å
−2 1
−1 2ã
, b =
Å
3
3
ã
, solve Ax = b using Cramer’s rule and verify
your answer is correct by checking whether Ax = b is satisfied.
2. Let A be a n × n matrix, prove the following three statements are all equivalent:
(a) Ax = 0 has nontrivial solutions (solutions other than 0).
(b) The determinant of A is zero.
(c) 0 is an eigenvalue of A.
3. Let A ∈ F
m×n
, m ≥ n with F = R or C be of full rank, prove that the
normal equation A∗Ax = A∗
b to the least squares problem min kAx − bk2
has a unique solution for any b ∈ F
n
.
