APPM 3310: Matrix Methods — Exam #2 — April 17, 2006
On the front of your bluebook print (1) your name, (2) your student ID number, and (3) a grading
table. Show all work in your bluebook. Textbooks, class notes and calculators are not permitted. If
you find that the arithmetic for this exam seems complicated, go back and check your work.
Please sign your bluebook under the Honor Code to indicate that you have neither given
nor received unauthorized assistance on this exam.
1. (40 points) For this problem, assume Ais a 3 ×4 matrix, except for part (a).
(a) State the complete (parts 1 and 2) Fundamental Theorem of Linear Algebra for a general
m×nmatrix A.
(b) Suppose you know that rng(A) = span
1
2
2
,
1
3
1
, find a basis for coker(A).
(c) Find an orthonormal basis for rng(A).
(d) Would Ax =bhave a solution if b= (1,1,1)T? What if b= (−4,1,1)T? Explain.
(e) Use your results from part (c) to find the QR decomposition for B=
1 1
2 3
2 1
.
2. (40 points) Each of the following unrelated questions involve some concept from class. The
concept is in bold-face. A complete answer for each part will include a definition of the bold-
faced word.
(a) Is the expression hu,vi=u1v1+u1v2+u2v1+v1v2an inner product for vectors in
R
2?
Why or why not?
(b) What are the L1and L∞norms on [0,1] for the function f(x) = (1/3) −x?
(c) Write the quadratic form q(x, y, z) = x2+ 4xy + 6y2−2xz + 9z2in the form q(x) = xTKx
for some symmetric matrix K. Is Kpositive definite? What does this tell you about the
extrema of q(x, y, z)?
(d) Show that if Qis an orthogonal matrix, then kQxk=kxkfor any vector x∈IRn, where
k·kdenotes the standard Euclidean norm.
3. (20 points) Consider the set of data given by: y= 1 at t=−2, y= 0 at t=−1, y= 3 at t= 3,
and y= 4 at t= 0.
(a) Use the method of least squares to find the best straight line y=α+βt for this data.
(b) Compute the error of this approximation.