Prof. A. Suciu
MTH 1230
LINEAR ALGEBRA
Spring 2001
Geometric interpretation of some linear transformations
1. Rotation-dilations
To interpret the linear transformation
T (x) =
a b
x
b
a
a
a
r cos
in polar coordinates:
=
. Then the matrix
b
b
r sin
geome
Resume Checklist
Accomplishment-Oriented
First Impression
Is the resume inviting to read, with clear sections and
ample white space?
Does the design look professional and not like a simple
typing job?
Is a qualifications summary included so the reader
imm
How to calculate seasonal
index
Pick time period (number of years)
Pick season period (month, quarter)
Calculate average price for season
Calculate average price over time
Divide season average by over time
average price x 100
Sioux Falls Cull Cow Prices
IE 7200
Supply Chain Engineering
Date:02/05/2017
Homework # 2
Problem 1.
Year
Month
Demand
(Units)
1
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Feb
Mar
100
125
140
160
185
210
225
235
215
200
190
180
205
230
240
Total error
2
3 week
moving
averag
Chap 2. The Geometry of LP
In the text, polyhedron is defined as P = cfw_ x Rn : Ax b . So some
of our earlier results should be taken with modifications.
Thm 2.1:
(a) The intersections of convex sets is convex.
(b) Every polyhedron is a convex set.
(c)
MS-E2140 Linear Programming
Exercise 7
Thu 06.10.2016
Maari-B
Week 4
This weeks homework https:/mycourses.aalto.fi/mod/folder/view.php?id=130935 is due no
later than Sunday 16.10.2016 23:55.
Exercise 7.1 Unboundedness and duality
Consider the LP:
c0 x
(P
MS-E2140 Linear Programming
Exercise 4
Fri 23.09.2016
Maari-B
Week 2
This weeks homework https:/mycourses.aalto.fi/mod/folder/view.php?id=130923 is due no
later than Sunday 02.10.2016 23:55.
Exercise 4.1 True and False Statements about Simplex
Course book
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Prof. A. Suciu
MTH 1230
LINEAR ALGEBRA
Spring 2001
SAMPLE EXAM 4
1. Let A =
13
. What is the area of the parallelogram spanned by the column vectors of 5I2 A?
26
2. Compute the area of the triangle with vertices at (1, 2), (5, 7), (3, 8).
3. Compute the a
Prof. Alexandru Suciu
MTH 1230
LINEAR ALGEBRA
Spring 2001
SOLUTIONS to EXAM 2
1. 12 points
12345
A = 1 0 1 2 3
23058
1 0 0 2 7
2
rref A = 0 1 0 3
5
0 0 1 0 1
2
(a) Find a basis for the image of A.
Columns of A corresponding to the pivot columns of rref A:
Prof. Alexandru Suciu
MTH 1230
LINEAR ALGEBRA
Spring 2001
EXAM 3
1. 10 pts Consider the independent vectors
1
1
1
1
v1 = , v2 = ,
1
1
1
1
0
0
v3 = .
1
1
Find an orthonormal basis cfw_w1 , w2 , w3 for the subspace of R4 which has cfw_v1 , v2 , v3 as a b
Prof. Alexandru Suciu
MTH 1230
LINEAR ALGEBRA
EXAM 2
1. 12 points
12
1 0
Let A =
23
(a) Find a basis for
3
1
0
the
45
2 3.
58
image of A.
(b) Find a basis for the kernel of A.
(c) Find the rank and the nullity of A.
Spring 2001
MTH 1230
Exam 2
2. 16 point
Prof. Alexandru Suciu
MTH 1230
LINEAR ALGEBRA
Spring 2001
EXAM 1
1. 10 points
(a) Use Gauss-Jordan elimination (rref) to solve the following system of equations. Carry out
the elimination all the way, before solving.
x1 + 2x2 + 3x3 + 4x4 = 5
2x1 + 5x2 + 7
Prof. Alexandru Suciu
MTH 1230
LINEAR ALGEBRA
Spring 2001
EXAM 4
1. 12 pts
(a) Compute the area of the region enclosed by the folowing quadrilateral:
1
4
4
2
3
2
2
1
2
0
2
1
(b) Compute the area of the parallelogram spanned by the vectors and .
1
1
3
1
Prof. Alexandru Suciu
MTH 1230
LINEAR ALGEBRA
Spring 2001
SOLUTIONS to EXAM 1
1. 10 points
(a) Use Gauss-Jordan elimination (rref) to solve the following system of equations. Carry out
the elimination all the way, before solving.
x1 + 2x2 + 3x3 + 4x4
2x1
Prof. A. Suciu
MTH 1230
LINEAR ALGEBRA
Spring 2001
SAMPLE EXAM 2
2
1
10
4
2
2 0
1. Let A =
6 3 0 1.
2
1 2 0
(a) Find a basis for im A.
(b) Find a basis for ker A.
(c) Find rank A.
1 34
2. Let A = 4 5 2.
1 3 8
(a) Determine whether the column vectors of A
1
2
3
4
5
6
7
8
Name:
NORTHEASTERN UNIVERSITY
DEPARTMENT OF MATHEMATICS
MTH 1230
FINAL EXAM
Spring 2001
Instructions: Put your name in the blanks above. Put your nal answers to each question in the
designated spaces. Calculators are permitted. A single sh
1
2
3
4
5
6
7
8
Name:
NORTHEASTERN UNIVERSITY
DEPARTMENT OF MATHEMATICS
MTH 1230
FINAL EXAM
Spring 2001
Instructions: Put your name in the blanks above. Put your nal answers to each question in the
designated spaces. Calculators are permitted. A single sh
iii) Material planning for subassembly A (level 1):
Since A is dependent on product 1, therefore the projected requirement for A in week 4 is 105
Lot size = PPB
Lead time = 1
Projected Requirement
Week
1
2
10
3
4
105
Scheduled Receipts
On hand at
end of w