Math 235: Linear Algebra
Midterm Exam 1
February 25, 2014
NAME (please print legibly): mill/firms?
Student ID Number:
CIRCLE YOUR INSTRUCTOR: Fatima Mahmood Geordie Richards
0 Read all instructions and all problems carefully.
o This is a closedbook an
MTH 235: Linear Algebra
Midterm 1
0 13
i) Lwbw
Indicate your instructor with a check in the box:
February
NAME (please print legibly):
Your University ID Number:
Giorgis Petridis
M ark Herman
c There are no notes, textbooks, etc. allowe
MTH 235 - Homework 3 Solutions
Question 1. This is done by modifying the proof of Theorem 1.9 as done in Example 6.
The set cfw_v1 , v2 is linearly independent as the vectors are not multiple of each other. Adding v3
gives rise to a linearly dependent se
MTH 235 - Homework 9 Sections 4.1 - 4.3
Please answer the questions in the order they are listed, STAPLE the pages together
and write clearly in the front of the assignment
COURSE ID NUMBER
Names of students you collaborated with.
Question 1. You are gi
MTH 235 - Homework 4 Section 2.1
Question 1.
Discussion: We know how T acts on (1, 2) and (1, 3), which incidentally form a basis for R2 , so we
express (1, 0) as a linear combination of the two vectors and use the linearity of T . For the second
part we
MATH 235
Final ANSWERS
May 5, 2015
1. (10 points)
Fix positive integers m, n and consider the vector space V of all m n matrices with entries
in the real numbers R.
(a) Find the dimension of V and prove your answer. Please carry out all the steps of your
MTH 235 - Homework 8 Section 3.3 & 3.4
Question 1. True of False?
Solution: (a) True Since Ax = 0 for A Mmn always has the trivial solution x = 0 Rn .
(b) False Consider the system of 2 equations in 3 unknowns x z = 0, y z = 0 which has
infinitely many so
MTH 235 - Homework 11 Sections 5.1, 5.2, 5.4
Question 1. True of False? For each of the following, decide if the statement is true or false.
Provide brief justification for your answer (you do not have to give full-blown proofs, brief explanations will su
MTH 235 - Homework 7 Sections 2.5, 3.1
Please answer the questions in the order they are listed, STAPLE the pages together
and write clearly in the front of the assignment
COURSE ID NUMBER
Names of students you collaborated with.
Question 1. True of Fal
MTH 235 - Homework 12 Chapter 6
Question 1. Let V be an inner product space.
(a) Prove that kx + yk2 + kx yk2 = 2kxk2 + 2kyk2 holds for all x, y V.
Solution:
kx + yk2 + kx yk2 = hx + y, x + yi + hx y, x yi
= hx, xi + hx, yi + hy, xi + hy, yi + hx, xi hx,
MTH 235 - Homework 9 Chapter 4
Please answer the questions in the order they are listed, STAPLE the pages together
and write clearly in the front of the assignment
COURSE ID NUMBER
Names of students you collaborated with.
Question 1. You are given two p
MTH 235 - Homework 6 Section 2.4-2.5
Question 1. True of False?
Solution: (a) True - by definition of invertibility.
(b) True - since IV IV = IV this follows from the definition of inverse map.
(c) False - If Onn is the n n zero matrix, then Onn A = Onn =
MTH 235 - Homework 6 Section 2.4
Please answer the questions in the order they are listed, STAPLE the pages together
and write clearly in the front of the assignment
COURSE ID NUMBER
Names of students you collaborated with.
Question 1. True of False? Fo
MATH 235
Midterm ANSWERS
April 14, 2015
1. (16 points)
(a) Find two disjoint finite spanning sets for the following subspace of R3 :
W = cfw_(x, y, z) R3 : x + y + z = 0.
Recall that two sets are disjoint when they have no element in common. Show all your
MATH 235
Midterm ANSWERS
February 26, 2015
1. (16 points)
(a) Find two disjoint finite spanning sets for the following subspace of R3 :
W = cfw_(x, y, z) R3 : x + y + z = 0.
Recall that two sets are disjoint when they have no element in common. Show all y
MTH 235 - Midterm 1 Review Questions
The following questions will be good practice for some of the questions you will
see on the midterm. However, doing these problems alone is not nearly enough.
You should also study HW1-HW3 and sections 1.1-1.6 in the t
MTH 235 - Homework 1 Sections 1.21.3
Please answer the questions in the order they are listed, staple the pages together
and write clearly in the front of the assignment
COURSE ID NUMBER
Names of students you collaborated with.
Question 1. For what valu
MTH 235 - Homework 1 Solutions
Question 1. Row reduction gives the following augmented matrix:
1 1 1
| t
| (t2 + 3t 10)/3
0 1 (t 4)/3
.
2
0 0 (t 6)(t 1)/3 | (t 1)(t + t 9)/3
When t = 6 there is no solution as the last row becomes (000|255/3).
When t = 1
MTH 235 - Homework 10 Chapter 4
Question 1. True of False?
(a) A square matrix A is invertible if and only if det(A) = 0. FALSE - a square matrix A is
invertible if and only if det(A) 6= 0.
(b) A square matrix A has linearly dependent columns if and only
MTH 235 - Homework 5 Sections 2.22.3
Question 1.
Discussion: To get [T ] we work column by column. The first column is [T (v1 )] : the coordinates
(with respect to ) of the image under T of the first basis vector in . The second column is
[T (v2 )] : the
Math 235 - Summer 2015
Homework 1
Due Wednesday May 27 in class
Remember: In this course, you must always show reasoning for your answers. You can use any result we have
proved in class, in textbook reading, or in a previous homework.
Problem 1
From Secti
MTH 235 - Homework 12 Chapter 6
This homework will not be graded
Question 1. Let V be an inner product space.
(a) Prove that kx + yk2 + kx yk2 = 2kxk2 + 2kyk2 holds for all x, y V.
(b) Suppose that x, y are orthogonal vectors. Prove that kxk2 + kyk2 = kx
MTH 235 - Homework 2 Sections 1.41.5
Please answer the questions in the order they are listed, staple the pages together
and write clearly in the front of the assignment
COURSE ID NUMBER
Names of students you collaborated with.
Question 1. You are given
Daniel Wong
Assignment 5 due 10/07/2016 at 06:00am EDT
fall16mth162
3. (1 point) Evaluate the indefinite integral.
1. (1 point) For each of the indefinite integrals below, select
which of the following trig substitutions would be most helpful
in evaluatin
Daniel Wong
MTH 162 Spring 2015
WeBWorK assignment number 3 is due : 09/23/2016 at 06:00am EDT.
The home page for the course contains the syllabus, grading policy, and other information.
Volumes and Work.
The primary purpose of WeBWorK is to let you know
Daniel Wong
MTH 162 Fall 2009
WeBWorK assignment number 10 is due : 11/11/2016 at 06:00am EST.
The home page for the course contains the syllabus, grading policy, and other information.
Series and integral test.
The primary purpose of WeBWorK is to let yo
Daniel Wong
Assignment 9 due 11/04/2016 at 06:00am EDT
fall16mth162
1. (1 point)
You are given the polar curve r = 3 cos().
2. (1 point)
Find the length of the spiral r = for 0 4.
Answer :
Solution:
Solution: We use the formula for the arc length. In this