Math 8 Homework #1 Solutions
1. Consider a mathematical theory in which a statement is any string of
letters where a letter is either a lowercase letter or a capital S. The only
axiom is: S. The logic works like this: in any theorem, any occurrence
of S c
Math 8 Homework #3 Solutions
1. For each statement below, decide if it is true or false. Justify your
choices. All x, y here are in R.
(a) cfw_1, 2, 3 = cfw_3, 1, 2
(b) cfw_1, 2, 3 = cfw_2, 1, 3, 3, 2
(c) cfw_5, = cfw_5
(d) cfw_5 cfw_5, 2
(e) cfw_1, 2
(f
Math 8 Homework #4
Due: Friday, April 29th in class
Instructions: I strongly recommend that you attempt all problems on
your own before consulting a classmate or myself. All assignments must be
written individually; no duplicates! Finally, neatness/presen
Math 8 Homework #5
Due: Friday, May 6th in class
Instructions: I strongly recommend that you attempt all problems on
your own before consulting a classmate or myself. All assignments must be
written individually; no duplicates! Finally, neatness/presentat
Math 8 Homework #2 Solutions
1. Give a direct proof of the following.
(1)
(2)
(3)
(4)
SP
P (G R)
G
P T
S T
Solution.
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(G R) P,
(G R) P,
G R,
P
S
T
S T
Contrapositive of (2).
De Morgans Law on (5).
By (3).
Modus Ponens with (6)
Math 8 Homework #4
Due: Friday, April 29th in class
Instructions: I strongly recommend that you attempt all problems on
your own before consulting a classmate or myself. All assignments must be
written individually; no duplicates! Finally, neatness/presen
Math 8 Homework
Not to be submitted, but do this as part of your test preparation.
Solutions will be on GauchoSpace by Tuesday.
1. Write out the addition and multiplication tables for Z9 .
Solution.
+ 0 1
0 0 1
1 1 2
2 2 3
3 3 4
4 4 5
5 5 6
6 6 7
7 7 8
8
Math 8 Homework #6
Due: Friday May 20th in class.
1. The following functions are from R to R. For each, determine whether
or not it is injective (AKA one-to-one) or surjective (AKA onto) (you
can draw graphs to figure out the answers, but pointing to a gr
Math 8 Homework #2
Due: Friday, April 15th in class
Instructions: I strongly recommend that you attempt all problems on
your own before consulting a classmate or myself. All assignments must be
written individually; no duplicates! Finally, neatness/presen
Math 8 Homework
Not to be submitted, but do this as part of your test preparation.
Solutions will be on GauchoSpace by Tuesday.
1. Write out the addition and multiplication tables for Z9
2. Calculate the following:
(a) 18 (mod 8).
(b) 2 (mod 9).
(c) 198 (
Math 8 Homework #1
Due: Friday, April 8th in class
Instructions: I strongly recommend that you attempt all problems on
your own before consulting a classmate or myself. All assignments must be
written individually; no duplicates! Finally, neatness/present
Math 8 Homework #5 Solutions
1. For each of the following relations, determine whether or not they are
reflexive, symmetric or transitive.
(a) Let | denote the relation on N where a|b if b is an integer multiple
of a, i.e., a divides b.
(b) Let be the rel
M108A Vector subspaces and linear transforms
In-class handout
Problems to solve and discuss:
Always F = R or C (think, say, that always F = R).
A useful method to establish that a set W is a vector space is to show that W V is a subspace of some bigger ve
108A EXAM 3
Take home 3 hors exam.
Notes:
(1)
(2)
(3)
(4)
This is a work-alone exam.
Write your solutions in a BLUE BOOK.
The exam is DUE AT THE START OF THE LAST LECTURE IN THE QUARTER.
Every problem costs 6 pts.
1.
(1) (3 pts) Let V be a vector space an
108A EXAM 1
In-class 75 min exam.
Notes:
(1) Lecture notes and/or other handwritten materials are allowed
(2) No textbooks, printouts, calculators, cells, internet access are allowed.
(3) Every problem costs 6 pts. Max for the exam is 42. Extra credit poi
M108A Basics: vector spaces, bases, linear (in)dependence, .
In-class handout
Problems to solve and discuss:
Always F = R or C (think, say, that always F = R).
1. Which sets V below form vector spaces? What are the zero vectors in the corresponding vector
108A EXAM 1
In-class 75 min exam.
Notes:
(1) Lecture notes and/or other handwritten materials are allowed
(2) No calculators, cells, internet access are allowed.
(3) Every problem costs 6 pts.
1. Answer yes/no. You do not have to prove your answers.
(1) A
Linear Algebra
Done Wrong
Sergei Treil
Department of Mathematics, Brown University
c Sergei Treil, 2004, 2009, 2011, 2014
Copyright
Preface
The title of the book sounds a bit mysterious. Why should anyone read this
book if it presents the subject in a wr
108A EXAM 2
In-class 75 min exam.
Notes:
(1) Lecture notes and/or other handwritten materials are allowed
(2) No calculators, cells, internet access are allowed.
(3) Every problem costs 6 pts, perfect score = 36.
1. You do not have to prove your answers i
108A EXAM 2 (PRACTICE VERSION)
In-class 75 min exam.
Notes:
(1) Lecture notes and/or other handwritten materials are allowed
(2) No calculators, cells, internet access are allowed.
(3) Every problem costs 6 pts.
1. You do not have to prove your answers in
M108A Rank Theorem
In-class handout
Problems to solve and discuss:
Always F = R or C (think, say, that always F = R).
1. Let B : V W be a linear transformation between abstract vector spaces. (Not necessarily a matrix!) State clearly the
definition of ran
M108A Invertibility and equations
In-class handout
Problems to solve and discuss:
Always F = R or C (think, say, that always F = R).
f
1. Recall M8 stuff: what is a bijection f between two sets, M N ? Give the definition.
A
2. Let V, W be vector spaces. I
M108A Determinants
In-class handout
Problems to solve and discuss:
Always F = R or C (think, say, that always F = R).
1. Find
det
x
.
y
2
2
2. Let A be an n n matrix, det A = D. For any number t what is det(tA)?
3. Compute
2
1
0
.
6
det 4
3
7
5.
.
n
The m