STA142B, S21
Prof. W. Polonik
Project 1
due Wed, April 20, 2022
A. Data analysis: Perform the data analysis projects as instructed in the file Project1.ipynb
that you can find in ’files’ on Canvas.
As instructed in the notebook, submit your project as an html file through upload to
GRADESCOPE. Also, submit the notebook and the images you used as zipped file through
CANVAS
B. Methodology: Answer the following questions, and submit your answers through
GRADESCOPE (either a picture or a scan – clearly readable).
So three different uploads in total. Also, plan ahead and try to not submit last minute
to avoid complications should unexpected technical problems occur.
1. (a) Show that for a given m-dimensional vector a, the map
fa(x) = 〈a, x〉
is linear.
(b) Now let A be a (n×m)-matrix. Show that the map
fA(x) = Ax
is linear.
2. Consider the symmetric matrix
A =
(
3
2
1
2 1
2
3
2
)
.
(a) Verify that A is PSD.
(b) i. Draw a figure that shows the points x1 =
(
1
0
)
and x2 =
(
1
1
)
and also their images
y1, y2 under the linear map induced by A, i.e.
y1 = Ax1 and y2 = Ax2.
ii. Find the unit eigenvectors of A and draw them into the same figure.
iii. Show how to obtain y1 and y2 in three steps by using the eigendecomposition
of A :
– projection onto unit eigenvectors
– scaling by eigenvalues and
– adding up the two scaled projections.
Do so by indicating the three steps in one figure.
iv. Find the image of the line passing through x1 and x2 under A and draw the
line and its image in a figure. Also include y1 and y2 in this figure.
