Northwestern University
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Math 230 Test #2 Solutions!
Fall Quarter 2012
Monday, November 19, 2012
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Write your NetID and your Student ID Number in the spaces provided at
the top of th
l) ( 20 pts.) Show ali your work
Lct A be a non-singular nxn matrix with n>l.
a) Calculate det(adjA) in terms of det A.
-\
b) Show that (a^)"1 = (det
-l
1) Let Wbe a subspace of R4 spanned by the vectors u = ( 3,-2,-1,0), v = (0,2,-2,0),
w = (3,0,0,-3) and z : (0,-4,0,4).
Determine
a) a basis for W,
b) dim W,
c) whether or not W: R4 .
Show all your steps explicitly.
0) IS enough 4-0 Qdorpé ltncaorhif Incl
1) ( 20 pts.) Show all your work
1 6 3 4
1 7 1 7
.
Let A =
1 8
1 10
1 0 15 0
a) Find the reduced row echelon form of A.
1
0
Reduced row echelon form of A is R =
0
0
0 15
1
2
0 0
0 0
0
0
.
1
0
b) Find a basis for the row space of A.
BR = cfw_(1,0,15,0
MATH 225
SOLUTIONS OF MIDTERM II
1
1
4 3
1 0 0
Question 1. (a) Let B = 0 1 2 and C = 9 2 0 . Compute the following
0
0 1
8 7 1
determinants:
det
1 12
B ,
2
det(3C 1 ),
det(B 19 C 3 ),
det
(2B)C
T
.
(b) Let A be a 3 3 matrix with det(A) = 1. Express the ma
June 15, 2012
MATH 230 HOMEWORK # 1
(Due: June 22, 2012, Friday)
(Your solutions will be collected in the class)
1. There are k boxes. Balls are placed at random one at a time into the boxes until, for the first
time, some box has two balls. Find the prob
February 12, 2013
MATH 230 HOMEWORK # 1
(Due: February 19, 2013, Tuesday)
( Submit your homework to my office till 5:00 pm, latest)
(Please, make sure that your name and department are written on the top of the first page of your homework)
1. If n distinc
4) (20 pts.) Show ali your work.
2
Given that yi (jt) - is a solution of 2x y" + 3xy' - y = O , x>0,
x
fmd the general solution by using reduction of order and identify the second
linearly independent solution. (Other solutions will not be accepted).
l',
Math 225
2008-2009 Fall
Final Exam Questions
1) a) Let A be an nxn matrix. Show that A can be written as the sum of a symmetric and a
skew symmetric matrices.
cfw_
b) Suppose u1 , u 2 , u 3 , u 4 , u 5 be a basis for R 5 . If c 2 , c 2 , c3 , c 4 , c5 ar
2) (20 pts.) Show ali your work.
Let * C =
e R2 and W =
- O be given.
a) Show that Wc is a subspace of R2.
use
LO O ]
be
n
o
M
O
II
a
9
ts a
d U, x
O
b) Using the definition given part a), fmd Wc n WD when C =
and
D=
l
-l
1
[H]
OA
P re
C O O J 6 LL> O U
L
I N E A R
A
L G E B R A
Erin P. J. Pearse
Rm = colsp(A) null(AT)
Rn = rowsp(A) null(A)
rowsp(A) = ran(AT)
rank(A) = r = dim rowsp(A)
Axp = b
colsp(A) = ran(A)
rank(A) = r = dim colsp(A)
Ax = b
xp
x = xp + xh
Axh = 0
b
A
null(A)
nullity(A) = n-r
0
xh
0
A
Math 220, Linear Algebra, Fall 2015
HW 1, Due Fri., Oct. 16, 15:30
Up to 2 (but NOT 3 or more) students can submit the same solution sheet
and get the same grade.
1. The trace Tr(A) of an n n matrix A = [aij ] is dened as the sum of its diagonal
n
entries
CS113IntroductiontoComputingforEngineers
Fall20142015ClassSyllabi
CourseContent
The students will be expected to be familiar with the following concepts, such as programming with
MATLAB, basic data manipulation, MATLAB user interface, scripts, vectors and
Nov-3, 2014
Bilkent University
Department of Industrial Engineering
IE 262
Homework 3
Due Date: November 17th, 2014 (Monday)
1) Acylindrical riser with diameter-to-length ratio = 1.0 is to be designed for a sand casting
mold. The casting geometry is illus
October-10, 2014
Bilkent University
Department of Industrial Engineering
IE 262
Homework 1
Due Date: October 17th, 2014 (Friday)
1) (40 pts) Please answer this question based on your own thoughts and ideas. You can use
internet sources, but you need to pr
IE 262
Homework Assignment 4
Due Date December 9th, 17:40
1) Low carbon steel having a shear strength of 220 N/mm2 is cut in a turning
operation with a cutting speed of 3.0 m/s. The uncut chip thickness is 0.20 mm and
the depth of cut is 3 mm. The rake an
COURSE PROJECT
Program in Cultures, Civilizations, and Ideas
Humanities 112: Modernity and Tradition
Spring 2015
Letter to Posterity: A Response
Due Date: March @ 21:00 via Moodle/Turnitin
After discovering Cicero's letters to his friend Atticus in the ca
5)( 20 pts.) Show ali your work.
Let y(^ - 5y" + 4y = (l + x)ex + 2e'x + sin3x + jccos3x be the given 4-th order nonhomogenous differential equation.
a) Find the general solution of the corresponding homogenous differential equation.
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