Description
1. Let z =
”
1
1
#
and w =
”
1
−1
#
.
a) Sketch the subspace spanned by z in R
2
.
b) Sketch the subspace spanned by w in R
2
.
c) Sketch span {z, w} in R
2
.
d) Are z and w orthogonal? Why or why not?
e) Do {z, w} form an orthonormal basis? Why or why not? If not, can you modify
z and w to form an orthonormal basis?
2. Consider the line in R
2 defined by the equation x2 = x1 + 1.
a) Sketch the line in R
2
.
b) Does this line define a subspace of R
2
? Why or why not?
3. You collect ratings of three space-related science fiction movies and two romance movies
from seven friends on a scale of 1-10.
Movie Jake Jennifer Jada Theo Ioan Bo Juanita
Star Trek 4 7 2 8 7 4 2
Pride and Prejudice 9 3 5 6 10 5 5
The Martian 4 8 3 7 6 4 1
Sense and Sensibility 9 2 6 5 9 5 4
Star Wars: Empire Strikes 4 9 2 8 7 4 1
You put this data into a matrix X (available in the file movie.mat) and decide to model
(approximate) as the product of a rank-r taste matrix with orthonormal columns and
a weight matrix. That is, X ≈ TW.
a) What is the rank of X? Relevant Python commands are
numpy.linalg.matrix rank().
b) What are the dimensions of T and W (in terms of r)?
c) You know that each user’s ratings have an average value that is greater than zero
because the scale is 1-10. And you suspect the baseline (average) rating may
differ from user to user. To account for this you decide your first basis vector in
the taste matrix should be
t1 =
1
√
5
1
1
.
.
.
1
Choose w1j so that each element of the vector t1w1j equals the average value j
th
column of X, denoted as X:,j . Find an expression for w1j that depends on t1 and
X:,j .
d) Define wT
1 =
w11 w12 · · · w17
and find the rank-1 approximation to X that
reflects the baseline ratings of each friend, t1wT
1
.
e) Which friend has the highest baseline rating? Which friend has the lowest baseline
rating?
f) Find the residual not modeled by t1wT
1
, that is, X − t1wT
1
.
Do you see any
patterns in the residual? Briefly describe them qualitatively.
This problem is continued in a homework assignment.
