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
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
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
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
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, c
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, c
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: Char
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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 argument
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
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 CLMSWZ
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
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 sta
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 closu
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
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, eigen
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 scala
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 answer
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 poi