Description
1) You collect ratings (out of 100) of three classes from four of your friends.
Bao: (CS760, 36), (ECE533, 40), (MATH521, 20)
Julia: (CS760, 72), (ECE533, 80), (MATH521, 40)
Vivek: (CS760, 90), (ECE533, 100), (MATH521, 50)
Jamal: (CS760, 54), (ECE533, 60), (MATH521, 30)
a) Express these ratings in a matrix X where the column indicates the friend (order
the columns as Bao, Julia, Vivek, Jamal), the row indicates the class (ordered as
CS760, ECE533, MATH521), and the value is the corresponding rating.
b) Suppose X is expressed as the outer product of a taste vector t and an affinity
weight vector w, that is, X = twT and assume t =
9
10
5
. Find w.
c) Your friend Brianna wasn’t able to complete your survey, but did rate ECE533
as 50. Assuming Brianna has the same taste profile as the rest of your friends,
what would her ratings for CS760 and Math521 be?
2) Suppose a 4 × 5 rating matrix X reflecting ratings of four movies by five people is
decomposed as the product of a rank-2 taste matrix T =
1 1
1 −1
1 −1
1 1
and rank-2
affinity weight matrix W =
”
8 6 4 5 4
2 3 3 −2 −1
#
, that is, X = TW.
a) Find X.
b) Write X = t1wT
1 + t2wT
2 where t1, t2, w1, w2 are column vectors and the first
elements of t1 and t2 are given by [t1]
1 = [t2]
1 = 1. Find one choice for t1, t2, w1
and w2.
3) Let X =
”
A B
c
T d
T
#
where A is 2 × 3, B is 2 × 2, c
T
is 1 × 3, and d
T
is 1 × 2.
a) Express the product R = XXT
in terms of A, B, c, d.
b) What are the dimensions of R?
4) a) Let y = Ax. Denote the columns of A as a1, a2, . . . , an. Express y as a weighted
sum of the columns of A.
b) Let A =
”
1 −1 2
1 1 0#
. Consider the columns of A as vectors in R
2
, and plot
them with the first element on the horizontal axis, and the second element on the
vertical axis.
c) Let y = Ax with x =
1
−1
1
and A as in part (b). Draw a picture to find y by
expressing it as a weighted sum of vectors you ploted in (b).