Homework 11: Due Friday 5th of December
1. Drop bears are large canivorous Australian marsupials, somewhat similar to koalas, that are known to attack campers by dropping onto tents
during the night.
(a) Suppose on any given night, all drop bears only dro
Homework 9: Due Friday 16th of November
d
1. Using moment generating functions, show that if Xn = Bin n, then
n
d
Xn = Po() as n .
2. Let X and Y be independent random variables, with known moment
generating functions MX (t) and MY (t) and Z be such that
MATH 309
Spring
2017
Practice Exam I =96?>398=
") Determine for which value of 2 the following homogeneous system has a
nontrivial solution.
2B" B# 0
6B" 2 B# !
A) " B " C # D # E $ F $ G % L % I & J) &
solution : To get a nontrivial solution we need the
MATH 309
Spring
PRACTICE EXAM I
") Determine for which value of 2 the following homogeneous system has a
nontrivial solution.
2B" B# 0
6B" 2 B# !
A) " B " C # D # E $ F $ G % L % I & J) &
-$
"
#
# "
!
"
#
%
"
$
#) Given E &
and F
#
#
$
% " #
then c#$
Review:
@" @# @: vectors in 8
-" -# -: scalars
-" @" -: @: is a linear combination of
@" @:
with weights -" -# -:
The set of all possible linear combinations of @" @: is called the span of these vectors:
[email protected]" @:
-" @" -: @: all possible weights
We can
EXAMPLE Chemical Reaction
We did this example before in class, but the second method connects to the recent
lecture on vector equations we can set up the problem as a system of linear equations
or aas a vector equation. If you watch how the problem gets s
Remember: We observed that for any two matrices E F with sizes so that EF is
defined:
row3 EF row3 E F
(we will use this again and again, below)
Theorem The multiplication IE (where I is an elementary matrix of the correct size)
performs on E the same ele
For learning (and exam) purposes, you should be able to give definitions of the following
terms.
Notes: 1) while a paraphrase of a definition is ok, it must be correct. Sometimes
students, when trying to paraphrase, actually misstate the definition making
MATH 309
Spring 201(
PRACTICE EXAM II - SOLUTIONS
1) Let V be the inner product space of 2 x 2 matrices with inner product given by
E F #+" ," +#" ,#" +"# ,"# #+# ,#
"
"
Find llFll where F
! "
A) 1 B) # G $ H # I & J ' K ( L ) M $
solution: llFll F F #
MATH 309
Spring 2017
PRACTICE EXAM II
1) Let V be the inner product space of 2 x 2 matrices with inner product given by
E F #+" ,"
+#" ,#"
+"# ,"#
#+# ,# .
"
"
! "
A) 1 B) # G $ H # I & J ' K ( L ) M $
-Find llFll
where
F
"
2) Using the inner produ
VERSION 1, with 2 PARTS (previously done): suppose X is a linear function, where
X 8 7 , with standard 7 8 matrix E
X is onto: that is, for every possible , 7
there is at least one B 8 for which
X B ,
For every possible , 7 , there is at
least one B 8 for
Mathematics 309
Exam 1
13 February 2013
Directions: This exam should consist of 10 multiple choice questions and 1 hand-graded
question. Multiple choice questions are worth 5 points apiece. The hand-graded question is
worth 10 points. The bonus problem wi
Math 309, Matrix Algebra
Exam 1, Fall 2012 Corrections Highlighted in Red
p1
NAME_
Circle your section:
Section 1 (11-12)
or
Section 2 (1-2)
Put all your work on the exam pages. None of the problems requires a lot of computation, so
there should be plenty
Math 361, Problem Set 2
September 17, 2010
Due: 9/13/10
1. (1.3.11) A bowl contains 16 chips, of which 6 are red, 7 are white and 3
are blue. If four chips are taken at random and without replacement, nd
the probability that
(a) each of the 4 chips is red
Math 361, Problem set 3
Due 9/20/10
1. (1.4.21) Suppose a fair 6-sided die is rolled 6 independent times. A match
occurs if side i is observed during the ith trial, i = 1, . . . , 6.
(a) What is the probability of at least one match during on the 6 rolls.
Test yourself
In the rref of a $ % matrix, what can you say about the number of leading "'s ?
(a) There must be 3 of them.
(b) There may be 0, 1, 2, or 3.
(c) There could be 4.
True or false:
_In a row reduction: if the original augmented
matrix has no co
Test yourself (not mentioned in class)
1.
a) If the variables are C @ A and > (and B ? 1 and D are constants)
then the equation is linear
BC '[email protected] 1DA )> #
b) If the variables are just B and D , and all the other symbols are constants, then
the equation i
True or False:
1. If a set contains fewer vectors than there are components in each vector, then the set is
linearly independent.
" #
# %
False: for example, consider
$
'
% )
2. The columns of E are linearly independent if the matrix equation EB ! has on
The following are all equivalent
for linear X 8 7
X is called onto if, for every , 7 there is at least one B 8 for which
X B ,
For every , 7 , the equation X B , has at least one solution
7
For every , , there is at least one B 8 for which EB ,
7
Every ve
W @" @3 @:
B" @" B3 @3 B: @: !
equivalent to
B"
@" @3 @: B3 !
B:
E
B !
W @" @3 @: lin. independent
if there is only the trivial solution
[email protected]" [email protected][email protected]: 0
W @" @3 @: lin. dependent
if there there are nontrivial solutions
that is, where not all wei
Vector equation for a line thru
< " # and ; " " ?
<
"
#
"
;
"
#
<;
"
#
#
>< ; >
line through ! and
"
"
"
#
< >< ;
>
#
"
same line translated by <
Example (see separate link in course syllabus):
another example of solutions in parametric vector
An observation about product of matrices (from last lecture, and important later in
lecture)
E
7 8 matrix
F
8 : matriB ," ,# ,:
+3"
+34
,"
+38 ,4"
,8"
-3"
row3 E F
" :
-3#
-34
,"#
, 4#
,8#
b"4
,44
,84
-38
row3 E F
" 8 8 :
,":
,4:
,8:
OLDER VERSION: suppose X is a linear function, where X 8 7 , with
standard matrix E
X is onto: that is, for every , 7
there is at least one B 8 for which
X B ,
For every , 7 , there is at least
one B 8 for which EB ,
(that is, the mapping B EB is onto
For
Product of Matrices:
For
E
F
7 8 matrix
8 : matriB ," ,# ,:
Then
G EF [E," E,#
E,8
Example
G
"
#
$
E
F
!
!
#
"
" #
%
#
!
"
#
% "%
" #
! #
"
#
$
#
#
"%
-$# "% +$" ,"# +$# ,#
Row-Column Rule for Computing EF
For E 7 8 and F 8 :, the8 -34 = elemen
Speedy One-Day Car Rental
Example
Everything that was on the pdf files is in the handout distributed (and linked in the
syllabus)
_
Matrix Terminology & Operations
+"
+#"
E
+3"
+"#
+#
+7"
+"
+3#
.
+"4
+#4
+34
+8#
+# .
. +"8
. +#8
.
+"4 .
+4 .
The ent