2
Homework Solutions
18.335 - Fall 2004
2.1
Count the number of oating point operations required to compute
the QR decomposition of an m-by-n matrix using (a) Householder
reectors (b) Givens rotations
RS Chapter 2 Random Variables
8/19/2015
Chapter 2
Random Variables
Random Variables
A random variable is a convenient way to express the elements
of as numbers rather than abstract elements of sets.
CS 61002 Lab#1
Spring 2016
(1) Working Copy Setup
The checkout commands done below put the checked out directories in your home directory. Put
the checked out directories in another directory if you p
Introduction to Statistics
1
Lecture 3
Outliers
An outlier is a data value that does not follow
the pattern of the other observations. Usually it
is too large or too small, compared with the other
val
Analysis I
Piotr Hajlasz
1
Measure theory
1.1
-algebra.
Definition. Let X be a set. A collection M of subsets of X is -algebra if M has the
following properties
1. X M;
2. A M = X \ A M;
3. A1 , A2 ,
Course Notes
for
MS4105
Linear Algebra 2
J. Kinsella
September 23, 2016
0-0
MS4105
Linear Algebra 2
0-1
Contents
I
Matrix Algebra
5
1 Matrices and Vectors
1.1
1.2
6
Properties of Matrices . . . . . .
Math 514 F15, Homework Assignment 5 - FD Methods for the Heat Equation
Due Friday, Dec. 4.
The homework can be submitted either in class or by e-mail by 11:59pm on the due date. Your
work must be comb
Math 514 F15, Exam 2 Review
Partial Differential Equations (PDEs)
Refer to Intro to PDEs notes.
1. Classification of PDEs
2. Be able to check that a given function is a solution to a given PDE.
Solvin
Math 514 F15, Homework Assignment 4 - Implicit Methods for ODE
Due Tuesday, Nov. 17.
The homework can be submitted either in class or by e-mail by 11:59pm on the due date. Your
work must be combined i
Math 514 F15, Homework Assignment 5 - FD Methods for the Heat Equation
Due Friday, Dec. 4.
The homework can be submitted either in class or by e-mail by 11:59pm on the due date. Your
work must be comb
Introduction to statistics
1
Lecture II
Organizing Data
A Pareto chart is a bar graph whose bars are
drawn in decreasing order of frequency or relative
frequency.
Definition 1. A side-by-side bar grap
8. the singular value decomposition
cmda 3606; mark embree
version of 5 May 2014
The singular value decomposition (SVD) is among the most
important and widely applicable matrix factorizations. It prov
CS 383C
CAM 383C/M 383E
Numerical Analysis: Linear Algebra
Fall 2008
Solutions to Homework 4
Lecturer: Inderjit Dhillon
Keywords: Householder Triangularization, Least Squares Problem
Date Due: Oct 1,
SVD computation example
T
Example: Find the SVD of A, U V , where A =
3 2 2
2 3 2
.
First we compute the singular values i by finding the eigenvalues of AAT .
17 8
T
AA =
.
8 17
T
The characteristic p
Chapter 1
Elements of Probability Distribution
Theory
1.1 Introductory Definitions
Statistics gives us methods to make inference about a population based on a random sample representing this populatio
Stochastic calculus for summable processes
1
Lecture I
Definition 1. Statistics is the science of
collecting, organizing, summarizing and analyzing
the information in order to draw conclusions.
It is
Chapter 2
Measure Spaces
2.1
Families of Sets
Definition 7 ( systems) A family of subsets F of is a system if,
Ak F for k = 1, 2 implies A1 A2 F.
A system is closed under finitely many intersections b
18.409 The Behavior of Algorithms in Practice
2/12/2
Lecture 2
Lecturer: Dan Spielman
Scribe: Steve Weis
Linear Algebra Review
A n x n matrix has n singular values. For a matrix A, the largest singula
Typos in Probability: Theory and Examples, 4th Edition
Contributions from Nate Eldredge, J.C. Li, Carl Mueller, Sebastien Roch, Byron Schmuland,
Antonio Sodre
Page numbers are those of the printed boo
Brad Nelson
Math 126
Homework #1
1/16/12
1 If u and v are m-vectors, the matrix A = I + uv is known as a rank-one perturbation of the identity. Show
that if A is non-singular, then its inverse has the
RS Chapter 1 Random Variables
Chapter 1
Probability Theory:
Introduction
Definitions Algebra
Definitions: Semiring
A collection of sets F is called a semiring if it satisfies:
F.
If A, B F, then A B
Orthogonal vectors and unitary matrices
Inner product
For any x1 , y1 , x2 , y2 , x, y Cm , , C,
(x1 + x2 ) y = x1 y + x2 y,
x (y1 + y2 ) = x y1 + x y2 ,
(x) (y) =
x y.
(That is, the inner product i
Tests and p-values
Suppose you have observations x1 , x2 , . . . , xn that come from some real world
process. You can interpret this as follows: x1 , x2 , . . . , xn are part of an infinite
sequence x
Lecture 4. The Singular Value
Decomposition
The singular value decomposition (SVD) is a matrix factorization whose computation is a step in many algorithms. Equally important is the use of the
SVD for