APPM 3310: Matrix Methods — Exam #2 — Summer 2012
On the front of your bluebook write (1) your name, (2) TEST 2/3310, (3) SUMMER 2012 and a grading
table. A correct answer with no supporting work may receive no credit while an incorrect answer with
some correct work may receive partial credit. Start each problem on a new page. No electronic devices of
any kind (e.g. cell phones, calculators, etc.) are permitted. A one-page sheet of notes is allowed. Justify
your answers and show all work!
Problem 1. (30 points) The following problems are unrelated:
(a) What is the definition of a positive definite matrix?
(b) If F > 0 and if B > 0 then is it true that F+B > 0? Why or why not?
(c) Let q(x) = xTMxfor any x∈Rnwhere M > 0, show that the expression
hx,yi ≡ 1
2[q(x+y)−q(x)−q(y)]
is the same as the inner product defined by M.
Problem 2. (30 points) The following problems are unrelated:
(a) Find a basis for all the four fundamental subspaces of the matrix F=
1+i2i−1
0 1 1
−2−2−2i1−i
.
(b) Show that every positive definite m×mmatrix Bis a Gram matrix (you must specify an inner
product and a set of vectors and show that Bis the associated Gram matrix.)
(c) Let k · kFand k ·kBbe two norms, does kvk ≡ 1
106 (kvkF+kvkB) define a norm? (Either explicitly
show that this satisfies all the properties of a norm or give an explicit example of when it fails to
satisfy one of the properties.)
Problem 3. (40 points) The following problems are unrelated:
(a) Find the closest vector in the space Sspanned by (1,1,0,0)Tand (0,0,1,1)Tto b= (3,1,2,1)T.
What is the distance of bfrom the space S? Show all work.
(b) Find the straight line that best fits the data {(ti, yi)}5
i=1 ={(1,1),(2,0),(3,−2),(4,−3),(5,−3)}
in the least squares sense.
(c) Find all values of aso that the vectors a
1and −a
1form an orthogonal basis of R2with
respect to the inner product defined by the matrix K=3 0
0 2 .
(d) Always True or False: Let Fand Bbe symmetric matrices, if xTFx=xTBxfor any x∈Rnthen
F=B. Justify.
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