Department of Mathematics 
University of San Francisco 
Mathematics 20201
Linear Algebra and Probability
Spring 2016
MWF 3:304:35, LS G12
Professor: Peter Pacheco
Office: Harney 540
Phone: 4226630
Email: domain: cs.usfca.edu, user: peter
Office Hours: MW 4:455:45, F 2:153:15, and by appointment
TA: Bryan Relampagos
Email: domain: dons.usfca.edu, user: barelampagos
Office Hours: W 121, Th 4:305:30 in HR 530 or HR 535
Class mailing list:
You will be automatically subscribed
to the class mailing list. Please note that this list uses
your USF email address. If you ordinarily read your
email using another account, be sure to forward your USF email
to the other account.
Course Syllabus (Here's a
PDF Version.)
Homework Assignments and Quizzes
Beware of the answers in the back of the book. There are mistakes.
 Quiz 1 Material.
 p. 8: 1 (The answer to 1c is incorrect in older printings:
it should be a plane. In newer printings the answer is correct.),
4, 9 (There are three possible fourth points, not four.), 16,
11, 17
 p. 19: 3, 4, 6, 12
 p. 29: 1, 2
Here's a key to one version
of quiz 1, and here's a key
to the other version.
 Quiz 2 Material.
 p. 29: 4 (find a solution that's not all zeroes), 6, 10
 p. 40: 9, 10, 11, 15, 16, 18, 26, 27
 p. 51: 1, 2, 3, 11, 12
Here are some solutions to the
problems that don't have answers in the back of the book.
Here's a key to one version
of quiz 2, and here's a key
to the other version.
 Quiz 3 Material.
 This version of
elimination uses a single n x (n+1)
"augmented" matrix to
store both the coefficients of the variables and the
righthand side. Rewrite the pseudocode so that it
uses an n x n matrix to store the coefficients
of the variables and an ndimensional column vector
to store the righthand side.
Here's a solution.
 This version of
back substitution uses an n x (n+1) augmented
matrix to store the matrix of coefficients and the righthand
side. Rewrite the pseudocode so that
it uses an n x n matrix of coefficients and an
ndimensional column vector to store both
the righthand side.
Here's a solution.
 p. 51: 6 (a singular system has no solution
or infinitely many solutions), 8, 14
Here's a key to one version
of quiz 3, and here's a key
to the other version.
 Additional Material for the first midterm
 Quiz 4 Material
 p. 63: 1, 3, 5, 9, 12, 14, 18
 p. 75: 2 (the order of the matrices is in number 1), 7ab, 9, 11,
15, 19, 32
 p. 89: 1
Here's a key to one version
of quiz 4, and here's a key
to the other version.
 Quiz 5 Material
 p. 89: 2 (how do you "undo" a swap? how do you undo a sequence
of swaps?), 14, 22, 27, 29
Here's a key to quiz 5.
 Quiz 6 Material. Note that quiz 6 will be on Wednesday, March 23.
 In an earlier quiz, we developed an
algorithm for forward substitution.
Determine the exact number of floating point operations carried out
by this algorithm. How does this compare the result we stated
in class for backward substitution? How many floating point
operations are
required in the forward substitution used in solving a linear
system using LUfactorization? Here's
a solution.
 p. 102: 4, 6, 10, 15.
Here's a key to one version
of quiz 6, and here's a key
to the other version.
 Quiz 7 Material.
 p. 115: 22, 24 (just PA = LU)
 p. 127: 1, 5 (M is the vector space of 2 x 2 matrices),
9a, 11, 15ab, 20, 23, 25.
Here's a key to one version
of quiz 7, and here's a key
to the other version.
 Additional Material for the Second Midterm
 p. 140: 4 (what are the free variables? what are the pivot
variables? find the special solutions and the complete solution);
6 and 8 (the matrices are in number 5); 10, 14, 16, 24
 p. 151: 7 (just Rx = 0)
 Quiz 8 Material.
 p. 151: 1, 13, 21 (a: just the nullspace)
 p. 163: 2, 4, 8, 12, 13abd, 14, 16, 18a (A^T is the
matrix obtained from A by interchanging the rows and columns
of A), 30
 p. 178: 2, 3, 11, 15, 18, 22
Here's a key.
Note: There are corrections to the originally
posted solutions to parts 1b and 1c.
 Quiz 9 Material.
 Here's a question on PageRank.
Here's a solution.
 p. 35 (in the
probability text): 1 (m(a) denotes the probability of
the simple event a), 2cde, 3cde, 7 (A with
a tilde above denotes the complement of the event A in the sample
space), 8, 9. Here are some
solutions.
 p.35: 21, 22, 23, 25, 31a (assume the students did not have
a flat tire). Here are some
solutions.
Here's a key to one version
of quiz 9, and here's a key
to the other version.
 Quiz 10 Material.
Here's a key to quiz 10.
 Quiz 11 Material.
 p. 113: 1acdfg, 2, 3, 12 (find the probability of
getting 3ofakind; find the probability of getting
two pair), 16. Here
are some solutions.
 p. 113: 1beh, 5 (here's a link
to the Dartmouth applets and some info on their use), 8,
10, 15, 18. Here
are some solutions.
Here's a key to one version
of quiz 11, and here's a key
to the other version.
 Quiz 12 Material.
Here's a key to one version
of quiz 12, and here's a key
to the other version.
 Additional material for the final exam:
 p. 150: 35, 39 (X and Y are independent if P(X=x and Y=y) =
P(X=x)P(Y=y), for all possible values of x and y.)
 p. 172: 1cd, 4abc, 7 (In number 7, suppose x is chosen at
random from the interval [0, 1]. Show that the events x > 1/3
and x < 2/3 are not independent.)
 p. 219: 25, 27
Here are some
solutions.
Code and PseudoCode
 Matrix operations
 Solving systems of linear equations
 Probability

Other Information
 A list of posssible topics for
the first midterm
 Solutions to one version of
Midterm 1. Solutions to the other
version of Midterm 1.
 A list of posssible topics for
the second midterm
 Solutions to one version of
Midterm 2. Solutions to the other
version of Midterm 2.
 This program
will draw histograms of numerical data.

This program will draw histograms of the binomial distribution.
 A list of additional topics
for the final exam.
Peter Pacheco
20160512