TEST 2 REVIEW
MATH 309, SECTION 6
You should remember the denitions and have a working knowledge of
the following concepts already covered: subspaces, linear independence, span,
basis, how to solve li
Test 1 Practice Test
Math 309, Section 6
1. Consider the following system of linear equations:
3y + z = 1
x + y 2z = 2
x 2y z = 3
Write the coecient matrix associated to the linear system. Use Gaussia
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Notes on Kernel/Image
0.1. Finding Kernel and Image of a matrix.
Let A M(m, n) be a matrix. The kernel of A (or nullspace)
Ker A = cfw_x Rn Ax = 0
is just the solution set to the system of homogeneous
SUPPLEMENT TO CHAPTER 3
1.1 Linear combinations and spanning sets
Consider the vector space R3 with the unit vectors e1 = (1, 0, 0), e2 = (0, 1, 0), e3 =
(0, 0, 1). Every vector v = (a, b, c) R3 can b
NOTES FOR 02/28/2011, 03/02/2011
MATH 309, SECTION 6
Note: We will not cover Chapter 4 in this class. Chapter 5 will be covered, but after covering some of
Chapter 6.
1. Coordinates (Chapter 3.6)
From
Math 309: Section 6 Reviews/Tests
Note that these do not cover the material from Chapters 7, 8.
1. True or False:
(a) If cfw_v1 , . . . , vn is a basis of V , and the linear map T : V W is an isomorp
MATH 309: SOLUTIONS TO REVIEW PROBLEMS FOR EXAM 3
Note: Be sure to also review Homeworks 6, 7 and 8.
(1) For each function f given below, check whether f has each of these four properties: linear, one
MATH 309: REVIEW PROBLEMS FOR EXAM 3
Note: Be sure to also review Homeworks 6, 7 and 8.
(1) For each function f given below, check whether f has each of these four properties: linear, one-to-one, onto
SOLUTIONS FOR EXAM 2
(1) Suppose r (u + 3v) + s (2u v) = 0. This implies (r + 2s) u + (3r
0. As fu; vg is linearly independent, we must have
s) v =
r + 2s = 0;
3r s = 0:
This yields easily r = s = 0.
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Each 100
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Math 309 Exam 2 Review
Exam 2 will be given in class, Friday Feb 21. The exam covers all the material done so far in the course,
which corresponds to Sections 1.11.6, 2.1 and 2.2, and 3.4 in the textb
Math 309 Exam 2 Review
Exam 2 will be given in class, Friday Feb 21. The exam covers all the material done so far in the course,
which corresponds to Sections 1.11.6, 2.1 and 2.2, and 3.4 in the textb
Day 29
Review for Exam 3
Exam 3 will be given in class, Wednesday March 26. The exam covers all the material done so far
in the course, which corresponds to Sections 1.11.6, 2.1 and 2.2, 3.4, 2.32.5,
Math 309 Exam 3 Review
Exam 3 will be given in class, Wednesday March 26. The exam covers all the material done so far in the
course, which corresponds to Sections 1.11.6, 2.1 and 2.2, 3.4, 2.32.5, 3.
SOLUTIONS FOR EXAM 3
(1) Suppose T : V ! W is one-to-one. For any v 2N (T ) we have T (v) =
0W = T (0V ). Thus v = 0V . Therefore N (T ) = f0V g.
Suppose N (T ) = f0V g. If T (v1 ) = T (v2 ), then T (
MATH 309: REVIEW PROBLEMS FOR EXAM 1
(1) Use the Gauss-Jordan algorithm to answer the following questions.
(a) Determine the conditions on a, b and c so that the following linear system
has at least o