Math 353/DSC 301, Spring 2019, Final Exam

Posted 2021-11-28 nameptyhon

Math 353/DSC 301, Spring 2019, Final Exam
Please remember to write down your name and student ID. This is the take-home portion which
has four problems. Only independently-finished and fully-justified answer will receive full credit.
1. [11 = 3 + 2 + 4 + 2 points] Consider the least-squares problem minxkb ? Axk2 where
Hint: the 3rd column of A is the sum of its first 2 columns and that the first 2 columns are
orthogonal.
(a) What is the range of A? What is the rank of A?
(b) What is the null-space of A?
(c) Find all least-squares solutions to the problem, and show that there is a least-squares
solution to the problem whose last component is zero.
(d) Find the least-squares solution to the problem which has the smallest 2-norm.
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2. [6 = 2 + 4 points] State the definition of the vector norms, then prove that for a matrix A ∈ Rn×n,
we have that
kAk1 = maxj∈{1,...,n}Xni=1
|aij |, kAk∞ = maxi∈{1,...,n}Xnj=1|aij |
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3. [6 = 3 + 3 points] Describe your understanding of the modified Gram-Schmidt and classical
Gram-Schmidt algorithms by focusing on the similarity and differences in terms of theoretical
operations, coding, and practical results. Then describe how different they would perform when,
e.g. they are applied to matrices such as the Vandermonde matrix. Hint: We did this test in
class.
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4. [6 = 3+3 points] Generate by yourself a nonsingular 4 × 4 matrix A with no zero entries and no
orthogonal columns.
a. Produce 3 Householder (orthonormal) matrices H1 H2, and H3 so that H3H2H1A is upper
triangular. Explain clearly how you generated these three matrices.
b. What is the QR decomposition from part a?

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