Analytical Methods for Applied Mathematics and Statistics
AMS 510

Fall 2015
Homework 3
AMS 510, Fall 2015
Assigned: Thursday, September 17
Due: Thursday, September 24 (beginning of class)
Objectives
To reinforce the concepts in LA4, LA5, and supplemental readings.
To practice skills with vector spaces, subspaces, linear (in)dep
Analytical Methods for Applied Mathematics and Statistics
AMS 510

Fall 2015
Homework 7
AMS 510, Fall 2015
Assigned: Thursday, October 29
Due: Thursday, November 5 (beginning of class)
Objectives
To practice and reinforce skills and concepts related to functions of multiple variables (limits, continuity, etc.), partial derivative
Analytical Methods for Applied Mathematics and Statistics
AMS 510

Fall 2015
Homework 4
AMS 510, Fall 2015
Assigned: Thursday, September 24
Due: Thursday, October 8 (beginning of class)
Objectives
To reinforce the concepts in LA6 (with parts of LA4 and LA5), LA7, LA9, and supplemental readings.
To practice skills with matrix rep
Analytical Methods for Applied Mathematics and Statistics
AMS 510

Fall 2011
AMS510 Analytical Method for AMS Final Exam
You are encouraged to do all the problems, the best 12 will be counted (1). Find the limit lim ( n + 2  2 n + 1 + n)
n
Solution:
n
lim
n + 2  2 n + 1 + n = lim [( n + 2  n + 1)  ( n + 1  n)]
n
1 1  n n+2
Analytical Methods for Applied Mathematics and Statistics
AMS 510

Fall 2014
AMS 510 Analytical Methods for Applied
Mathematics and Statistics
14 Fall Sample Final Exam
Name_ Student ID_
Total Score_
There are total of 13 regular problems and 1 extra
credit problem. Good Luck!
1
(1) (15 points) Given the matrix
1 0 2
A = 2 1 3 ,
4
Analytical Methods for Applied Mathematics and Statistics
AMS 510

Fall 2010
AMS 510 HW9 Solution
4.83 1 2 u= 2 , v = 3 , w =mu nv 3 1 (a) w =3 u1 u2 (b) Impossible 121 (c) 2 3 k = 0 k =11 / 5 314
(d)
12a 2 3 b =0 7a 5b c =0 31c
4.84 Expand f = xp 1 yp2 zp 3 , we can get x =a , y =2 a b , z =a bc . 4.87 For any u span S , there
Analytical Methods For Applied Mathematics And Statistics
AMS 510

Fall 2013
AMS510 Solution for HW5
Oct 28 2013
4.69
1
a) ba < ln b ln a < ba 1 < ln bln a < a Let f (x) = ln(x) f (x) = 1/x
b
a
b
ba
f (b)f (a)
1
Using mean value theorem: ba = f ( ), a < < b. 1 < f ( ) < a .
b
1
1
b) 6 < ln 1.2 < 5 is true when a = 5, b = 6
4.75
(
Analytical Methods For Applied Mathematics And Statistics
AMS 510

Fall 2014
AMS 510 Analytical Methods for Applied
Mathematics and Statistics
2014 Fall Midterm Exam I solutions
(1). (15 points) Given the linear mapping F : R3 R3 dened by
F (x, y, z) = (x + 2y z, y + z, x + y 2z)
(a). Find the matrix representation of F = Av, wher
Analytical Methods For Applied Mathematics And Statistics
AMS 510

Fall 2014
Lecture Notes
1
Lecture 1
1. Sets: notation
By listing the elements
S = cfw_x, y, z, . . .
or by giving properties
S = cfw_xwhere x has following properties:.
Union: S T all elements in either set
Intersection S T all common elements
Set dierence: S \ T
Analytical Methods For Applied Mathematics And Statistics
AMS 510

Fall 2014
Lower and upper triangular matrices. The inverse of a lower triangular
matrix is another lower triangular matrix. The inverse of an upper
triangular matrix is another upper triangular matrix.
Gauss elimination can be expressed by the multiplication of a
Analytical Methods For Applied Mathematics And Statistics
AMS 510

Fall 2013
Page 1
Theorems for differentiable functions
Rolle's Theorem
Page 2
Mean Value Theorem
Page 3
Cauchy Generalization Mean Value Theorem
Page 4
L'Hopital's Rule:
Analytical Methods for Applied Mathematics and Statistics
AMS 510

Spring 2014
Solutions to Homework 6: Vector Calculus
AMS 510, Fall 2016
Problems (100 points)
1. Let A, B : R3 R3 and let U : R3 R. Prove the following identities.
(a) (A B) = B ( A) A ( B).
(b) (U ) = 0. State any conditions on U that are necessary for this identity
Analytical Methods For Applied Mathematics And Statistics
AMS 510

Fall 2013
Page 1
Page 2
continuity of a function on an interval:
A function is continuous on an interval if it is continuous at every point on the interval
Page 3
Calculus
Page 4
Page 5
Page 6
Analytical Methods For Applied Mathematics And Statistics
AMS 510

Fall 2013
Page 1
Homogeneous equation
Ax=0
Row rank of A = row rank of M (where M is A augmented with a column of zeros)
Vector space:
S = cfw_v1,.,vn (can also be infinite set) is called a space when
Page 2
Some of the given vectors may be linearly dependent, in
Analytical Methods For Applied Mathematics And Statistics
AMS 510

Fall 2013
Page 1
System of linear equations:
3) Each pivot is equal to 1
4) Each pivot is the only nonzero entry in its column
Page 2
Page 3
Definition: two nonzero vectors, a and b, are linearly independent if
Definition:
Let v1, ., vn be n vectors
v1,.,vn a
Analytical Methods For Applied Mathematics And Statistics
AMS 510

Fall 2013
Page 1
Review
Column space: if b is a member of the column space, then there exists a solution. If b is
not in the column space of the matrix A, there is no solution
Look up definition for null space
dim(row space) + dim(null space) = n ?
Solution of a sy
Analytical Methods For Applied Mathematics And Statistics
AMS 510

Fall 2013
Page 1
Calculus
Chapter 1:
Set: collection of elements
number set: collection of numbers
Proof by induction example
Page 2
Chapter 2: series, sequences and limits of series
Sequences:
Page 3
Convergent sequence:
Upper bound for a sequence:
If a sequence
Analytical Methods For Applied Mathematics And Statistics
AMS 510

Fall 2013
Page 1
1) no solution: rank(A) < rank(M)
2) infinitely many solutions rank(A) = rank(M) < n
3) unique solution: rank(A) = rank(M) = n
Page 2
Page 3
LU Decomposition
A vector space is a set of vectors
but
A set of vectors is not necessarily
a vector space
Analytical Methods For Applied Mathematics And Statistics
AMS 510

Fall 2013
Page 1
Page 2
Page 3
Vocabulary:
Transpose of A
makes the columns into rows and rows into columns
Trace of a matrix: sum of the diagonal elements, tr(A) = trace of matrix A =
Identity Matrix:
Page 4
Diagonal Matrix:
Triangular matrix
Orthogonal matrix
Pag
Analytical Methods For Applied Mathematics And Statistics
AMS 510

Fall 2013
Page 1
If you feel grading was unfair, go to the TA first then Dr. Li will give a final judgement
2 midterms, 100 pts total, final exam is 200 pts, 200 pts for 8 home works
Page 2
Page 3
Page 4
Analytical Methods For Applied Mathematics And Statistics
AMS 510

Fall 2013
Page 1
Page 2
Monotone function: STRICTLY increasing or decreasing for this class
Functions to remember graphs for this class: x^2, sin(x), exp(x), x^3, x, ln(x), arcsin(x),
arccos(x)
Polynomial: algebraic functions
exponentials, trig, logs: transcendent
Analytical Methods for Applied Mathematics and Statistics
AMS 510

Spring 2014
AMS 572 Data Analysis I
Inference on two population means and two
population variances
PeiFen Kuan
Applied Math and Stats, Stony Brook University
AMS 572
PF.Kuan
1
Two population setting
I
Single crosssectional sample, comparing two subsamples
I
Compar
Analytical Methods for Applied Mathematics and Statistics
AMS 510

Spring 2014
Homework 2: Matrix Properties and Algorithms
AMS 510, Fall 2016
Easy Problems (36 points)
E.1. Do problem LA3.6 for the system
16x + ay = 2,
2ax + 2y = b.
Solution: The solution strategy follows that of the book.
(a) a R \ cfw_4, 4.
(b) (a, b) = (4, 1) or
Analytical Methods for Applied Mathematics and Statistics
AMS 510

Spring 2014
Solutions to Homework 4: Analysis
AMS 510, Fall 2016
Easy Problems (26 points)
E.1. Problem AC1.2.5.
n2 sin(n)
as n goes to infinity is 0. Proof:
n3 + 1
1
Let > 0 and choose N = . Then, for n > N ,
2
2
2
n sin(n)
n sin(n)
n2
1
1
n
0
=
<
= .
= <
n3