Math 1180 Assignment 1
1. Write each of the following systems in matrix form Ax = b and nd the general
solution.
(a)
x + y 2z = 5
2x + 3y + 4z = 2
Solution:
The matrix form of the equation is
1 1 2
23
4
x
y =
z
5
2
The augmented matrix is
1 1 2
23
4
whic
1
6.1.3: Not a linear vector space, axioms 5 and 6 fail to hold.
6.1.5: Not a linear vector space, axiom 8 fails to hold for c 1, d 1
1
1
0 , v 0
6.1.27: Not a subspace. Condition a) of Theorem 6.2 fails to hold for u
1
1
6.1.33: Is a subspace.
6.1.36:
1
MATH 1180
Midterm Exam II Sample
Spring 2010
Instructor: David Swigon
Problem 1: (20 points) Find all real values of k for which the matrix A is diagonalizable.
1 1 k
A 1 1 k
1 1 k
Problem 2: (15 points) Compute det A:
1 1
2 5
A
2 6
2 0
3
6
2 6
1 3
0
1
MATH 1180
Final Exam
Spring 2010
Instructor: David Swigon
NAME: _
Justify all your results.
In proofs justify every step.
If you run out of space, continue on the back of the page.
Absolutely no use of tables, notes, books, headphones, cell phones, calc
1
MATH 1180
Midterm Exam I Sample
Spring 2010
Instructor: David Swigon
NAME: _
Show all your work. Unsupported answers are treated as lucky guesses and will not be credited.
If you run out of space, continue on the back of the page.
Absolutely no use of t
1
MATH 1180
Midterm Exam I Sample
Spring 2010
Instructor: David Swigon
NAME: _
Show all your work. Unsupported answers are treated as lucky guesses and will not be credited.
If you run out of space, continue on the back of the page.
Absolutely no use of t
1
MATH 1180
Midterm Exam II Sample
Spring 2010
Instructor: David Swigon
Problem 1: (20 points) Find all real values of k for which the matrix A is diagonalizable.
1 1 k
A 1 1 k
1 1 k
Solution: Characteristic polynomial:
1
det( A I )
1
1
1
1
1
k
k (1
lhmo* ort #a so/n
P -?-z
ltf?r
/80
6
l-z o 71o o t-3/s
rc
da
cr-d
4 k,+P,5;
rD.,os ;de(Ael
Ez-tK,
d.L/#4
drbr"
ret4
A/t(
gl" cY
-4t.-, K,
La q J n,tq+K, L o cfw_J3k:-(_o
oro
7 olo
O o oc
OO\2
6a Q Jt
0o
3K
to
I =J
3
ol3
o
Oool
103-Vo
O / J-o/
m-nt4
/o7-Y
Uniqueness of Reduced Row Echelon Form
Many introductory linear algebra books either fail to mention this result, omit its
proof, or present a proof which is unnecessarily complicated or uses arguments beyond the
context in which the result occurs. Heres
Math 1180 Topics
For the Midterm Exam
Chapter 1
Concepts. You are expected to know and be able to operate comfortably with the following
concepts:
Vectors in Rn
Finding vectors coordinates (components)
Addition, scalar multiplication, subtraction of vecto
Quiz 1
Summer 2011
Math 1180 Your name: M i i O 1/1 S
pmééel/M 1.2 #22
1. [5 points] Determine whether the angle between the vectors
«w 1
1 2 . .
_1 and 3 IS acute, obtuse 0r rlght.
1 4 '
The CLMSWZV depemds 0m 7%? gag/b:
0f (08 9 , W'er (925 Cabl {I
_
Elementary Matrices
February 1, 2013
Denition 1. A matrix obtained by applying a single elementary row operation
to an identity matrix is an elementary matrix.
Lemma 2. Any elementary row operation on an m n matrix A can be performed by left multiplying A
Math 1180
Solutions to Homework 3
February 15, 2013
LADW Chapter 1 Number 3.2
Reection across the line x1 = x2 interchanges the two standard basis vectors:
T e1 = e2 =
0
1
,
T e2 = e1 =
1
0
so the standard matrix of T is
[T e1 T e2 ] =
0
1
1
.
0
LADW Chap
Math 1180 Assignment 4 Solutions
1. Let V and W be subspaces of a vector space U .
(a) Show that V W is a subspace of U .
Solution:
V W is non-empty, since it contains the zero vector of U .
For closure under addition, let u1 , u2 V W . Since u1 and u2 ar
Math 1180
Solutions to Homework 5
February 24, 2013
LADW Chapter 2 Number 8.2
a) Since we have four vectors in a four dimensional space, it is only necessary
to check linear independence. Suppose
0
0
0
0
1
1 0
3
1
2
(1)
c1 + c2 + c3 + c4 = .
0 0
2
3
1
1
Math 1180
Solutions to Homework 6
March 29, 2013
LADW Chapter 4 Number 1.2
a) With A =
4
2
5
,
3
det(A I ) = det
4
2
5
= 2 2 = ( 2)( + 1)
3
so the eigenvalues are 2 and 1.
For the eigenvalue 2, eigenvectors are non-zero members of the null space
of
2 5
A
Math 1180
Solutions to Homework 7
April 10, 2013
LADW Chapter 5, Number 1.8
(a) Taking v = x gives
x
2
= x, x = 0,
so x = 0.
(b) Let v V . Since cfw_v1 , . . . , vn is a spanning set, there are scalars c1 , . . . , cn
such that v =
cj vj . Then
x, v = x,
Math 1180 Midterm Exam Rewrite
Due by noon, March 1, 2013
Name:
Instructions: No books or notes may be used during the exam. Attempt all problems, giving complete and clear explanations of your answers.
1. (10 points) Determine whether the following syste
Math 1180 Midterm Exam Solutions
February 25, 2013
Instructions: No books or notes may be used during the exam. Attempt all problems, giving complete and clear explanations of your answers.
1. (10 points) Determine whether the following system is consiste