Accounting and the
Calculus II: 21 241
Introduction
Business
Environment
Section D and
A Rectitations
Chapter 1
What do businesses do?
Why do we need accounting?
Why Is Accounting Important?
Accounting is the
information system that
measures business
acti

M241: Matrices and Linear Transformations
Midterm 2 Practice Problems
Textbook. The answers to all of these problems are at the end of the book.
Section 2.3 Problems 21, 23, 29, 31, 33, 35.
Section 2.4 Problems 29, 37.
Section 2.6 Problems 25, 29, 37,

M241: Matrix Algebra
Midterm 3 Practice Problems
Textbook. The answers to some problems in Section 4.3 are included in the posted solutions. The
answers to most of the other problems are at the end of the book.
Section 3.3 Problems 3, 7, 11, 19.
Section

21-241
Matrices and Linear Transformations
D. Handron
Exam #2 Review, Spring 2015
1. Let
2
0
A=
0
0
0
3
0
0
0
1
3
0
0
0
0
2
Find all the eigenvalues of the matrix A. For each eigenvalue, express the eigenspace as a
span of vectors.
2. Let
1 1 2
A= 1 2

Jason Starr
Fall 2005
18.01 Calculus
Due by 2:00pm sharp
Friday, Oct. 14, 2005
Problem Set 3
Late homework policy. Late work will be accepted only with a medical note or for another
Instituteapproved reason.
Cooperation policy. You are encouraged to work

18.01 Calculus
Due by 2:00pm sharp
Friday, Dec. 2, 2005
Jason Starr
Fall 2005
Solutions to Problem Set 7
Late homework policy. Late work will be accepted only with a medical note or for another
Instituteapproved reason.
Cooperation policy. You are encoura

Jason Starr
Fall 2005
18.01 Calculus
Due by 2:00pm sharp
Friday, Sept. 16, 2005
Solutions to Problem Set 1
Part I/Part II
Part I(20 points)
(a) (2 points)
(b) (2 points)
(c) (2 points)
(d) (2 points)
(e) (2 points)
(f ) (2 points)
(g) (2 points)
(h) (2 po

Description
Linear algebra covers
material which is essential
to anyone who does any
mathematical
computation in
engineering and the
sciences. In application
and in class, the subject
divides naturally into two
parts: computation and
formal structure.
The

Assignments: Week #1
Monday: Sections 2.1.
Wednesday: Section 2.2.
Exercises:
Reading:
Friday: Sections 2.1 and 2.2.
To find the exercises for Week #1, first go read
the statement regardingacademic integrity on the
Policies page, then click the link to t

M241: Matrices and Linear Transformations
Midterm 1 Practice Problems Answers
Textbook. The answers to all these problems are at the end of the book.
Problem 1. Solve the systems below:
x1 +2x2 +3x3 +5x4 = 2,
2x +4x2 +8x3 +12x4 = 6,
1
3x1 +6x2 +7x3 +13x

Math 241: Matrices and Linear Transformations
QUIZ 2 SOLUTIONS
Sections F and G
Section F Answers
1. Let the function T : P2 R3 be dened by
p(0)
T (p(t) = p(1) .
p(2)
(a) (5 pts.) Use the denition of a linear transformation to prove that the T is a
linear

Math 241: Matrices and Linear Transformations
QUIZ 3 SOLUTIONS
Sections F, G
Section F Answers
1. (7 pts.) Let
1
1
0
0
0
3
x1 = , x2 = , x3 = .
1
2
0
0
0
4
Suppose that Gram-Shmidt was applied to x1 , x2 , x3 and the result was the orthonormal
vectors

M241: Matrices and Linear Transformations
Homework Assignment 8
Answers
Graded Problems: Section 3.4 Problems 12, 16, 28, 32; Section 4.2 Problem 4; Problem
A.
Section 3.4
Problem 12. If a2 ma1 is orthogonal to a1 , then aT (a2 ma1 ) = 0, thus aT a2 = maT

M241: Matrices and Linear Transformations
Midterm 3 Practice Problems Answers
Textbook. Here are the solutions to the textbook problems that do not have answers at the end
of the textbook.
Section 4.3.
Problem 14. The columns are linearly dependent if and

M241: Matrices and Linear Transformations
Midterm 1 Practice Problems
Textbook. The answers to all these problems are at the end of the book.
Section 1.3 Problems 23, 31.
Section 1.4 Problems 1, 11, 13, 41.
Section 1.5 Problems 15, 27, 29.
Section 1.6

M241: Matrices and Linear Transformations
Midterm 2 Practice Problems Answers
Textbook. The answers to all of these problems are at the end of the book.
Problem 1. We need to prove that
S(c1 w1 + c2 w2 ) = c1 S(w1 ) + c2 S(w2 )
for any w in W .
Let v1 = S

Math 241: Matrices and Linear Transformations
QUIZ 1
Section F (9:30 10:20 AM)
You must show all your work! Closed books, closed notes. No calculator.
Duration: 30 min.
Your Name: SOLUTIONS
1. Find the A = LDU factorization for the matrix
1
2 1 3
A = 1 4

M241: Matrices and Linear Transformations
Homework Assignment 9
Answers
Graded Problems: Section 5.2 Problem 2, 10; Problem A (graded as four problems).
Section 5.1
Problem 14. Note that rank(A) = 1, this means that 1 = 0 is an eigenvalue with geometric
m

Math 241: Matrices and Linear Transformations
Fall 2014 Tentative Syllabus
Lecturer:
Dr. Irina Gheorghiciuc
Oce: Wean Hall 8126
Website: www.cmu.edu/blackboard
Oce Hours: M 10:30 11:30 PM, W 1:30 2:30 AM
Phone: (412) 268-3023
E-mail: [email protected]

Review Questions:
1. What is a linear system of
equations?
2. What are the three
operations we can perform on a
system?
3. How many solutions
might an system have?
4. What are the leading
variables of a system? The free
variables? How do we decide
which a

21-241
Matrices and Linear Transformations
D. Handron
Exam #1 Review
Spring 2015
1. Find all solutions to the linear system of equations
x1 + 2x2 + 4x3
= 5
x1
+ 2x3 2x4 = 7
x2 3x3 + x4 = 6.
Express the solution set in vector form.
4
2. Is the vector 2 a

1. What is a real vector
Review Questions:
space? A complex vector
space?
2. What do linear
transformations have to do with
linear combinations
3. Why is differentiation a
linear transformation?
4. What are two linear
transformations that can be
defined f