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
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
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 '?@ 1DA )> #
b) If the variables are just B and D , and all the other symbols are constants, then
the equation i
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
System of Linear Equations
variables unknowns
B"
B#
B8
+" B" +"# B# +"8 B8 ,"
+#" B" +# B# +#8 B8 ,#
+3" B" +3# B# +38 B8 ,3
+7" B" +7# B# +78 B8 ,7
constant coefficients (real or complex
numbers)
A solution means a list of numbers
B " c"
B 8 c8
such th
Ma 309: Matrix Algebra
Model Solutions to Homework Assignment 12
Prof. Wickerhauser
Exercise
1 of Section 6.7, p.403.
a. Leading principal submatrices are A1=2 and A2=A. Check
that det(A1)=2>0 and det(A2)=3>0. By Theorem 6.7.1, A is
positive definite.
b.
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
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
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:
Span@" @:
-" @" -: @: all possible weights
We can
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.
Math 361, Problem set 4
Due 9/27/10
1. (1.4.26) Person A tosses a coin and then person B rolls a die. This is repeated independently until a head or one of the numbers 1, 2, 3, 4 appears,
at which time the game is stopped. Person A wins with the head, and
Preliminary Information about Exam 2, Math 309, Fall 2015. This information will also be
posted as a link in the syllabus.
I expect any changes from what is written here to be only small ones. I will notify you if
necessary of changes as the exam is const
An Economic Example
The example illustrates how a system of linear equations might describe the functioning
(production and consumption) in a very simple economy. The thing to focus on here is
not the numbers but how the equations are set up to describe a
Review: Using the Invertible Matrix Theorem, explain why the matrix is or is not
invertible (it's good practice if you can see more than one reason)
"
!
E
!
!
#
'
!
!
$
"
#
!
%
$
%
#
"
!
E
!
!
#
'
!
!
$
"
#
"
%
$
%
#
#
E "
!
"
!
"
%
#
!
!
!
X % % where
Ma 309: Matrix Algebra
Model Solutions to Homework Assignment 9
Prof. Wickerhauser
Exercise
1bgik of Section 6.1, p.323.
Compute the characteristic polynomials p(v)=det(A-vI), then
find N(A-vI) for each root of p(v)=0 by reduction to reduced
row echelon f
Elementary Row Operations (EROs)
for a matrix (or a system of linear equations)
1) interchange any two equations (rows)
2) multiply any equation (row) by a nonzero
constant 3) add a multiple of one equation (row) to
another equation (row)
Each operation i
All of the projected pictures in Lecture 3 were from the textbook, so there's not much that I
felt I needed to post in this pdf file for you today.
True or False
a) In some cases a matrix may be row reduced to more than one matrix in reduced echelon
form
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
!@" !@3 !@: 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: