Versine - Wikipedia, the free encyclopedia
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Versine
From Wikipedia, the free encyclopedia
The versine or versed sine, versin(), is a trigonometric function equal to 1 cos() and 2sin2(). It
appeared in some of the earliest trigonometric tables a
Starting with Two Matrices
Gilbert Strang, Massachusetts Institute of Technology
Imagine that you have never seen matrices. On the principle that examples are amazingly powerful, we
study two matrices A and C . The reader is requested to be exceptionally
Orthogonality Tutorial
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ORTHOGONALITY TUTORIAL
TUTORIALS
TUTORIALS HOME
GENERAL MATH
NOTATION &
METHODS OF
PROOF
INDUCTION
COMPLEX
NUMBERS
POLYNOMIALS
Orthogonal Sets
Vectors v, u are orthogonal or perpendicular to each other if v
v
LINEAR ALGE
LECTURE 2
Orthogonal Vectors and
Matrices
OBJECTIVE: Orthogonality is central to many of
the main algorithms on linear algebra. We review the
ingredients: orthogonal vectors and matrices.
2-1
Transp ose
Denition
The transpose AT of an m n matrix A is n m
Chapter 04.02
Vectors
After reading this chapter, you should be able to:
1.
2.
3.
4.
5.
define a vector,
add and subtract vectors,
find linear combinations of vectors and their relationship to a set of equations,
explain what it means to have a linearly i
EE 505 B Fall 2011
Midterm Exam Solutions
1. [23 points]
(a) [3 points] b is in the column space of the matrix, or (equivalently) b is a
linear combination of the columns of A
(b) (i) [6 points] The eigenvalues are found using the characteristic equation:
EE 505 B Fall 2011
Midterm Exam
Nota bene: For full credit you must show all of the steps or reasoning that you used to
get the answer.
1. [23 points]
(a) [3 points] If one appends a column vector b to a matrix A, then the column
.
space of the matrix get
Edward Neuman Department of Mathematics Southern Illinois University at Carbondale [email protected]
One of the nice features of MATLAB is its ease of computations with vectors and matrices. In this tutorial the following topics are discussed: vectors a
Mathematics of the DFT
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Mathematics of the DFT
In the signal processing literature, it is common to write the DFT and its inverse in the more pure form
below, obtained by
List of trigonometric identities - Wikipedia, the free encyclopedia
Page 1 of 25
List of trigonometric identities
From Wikipedia, the free encyclopedia
In mathematics, trigonometric identities are
equalities that involve trigonometric functions and
are tr
List of integrals of trigonometric functions - Wikipedia, the free encyclopedia
Page 1 of 10
List of integrals of trigonometric functions
From Wikipedia, the free encyclopedia
The following is a list of integrals (antiderivative
functions) of trigonometri
The Eigenvalue Problems - 8.8
1. Definition of Eigenvalues and Eigenvectors:
Let A be an n ! n matrix. A scalar ! is said to be an eigenvalue of A if the linear system Av " !v has a
nonzero solution vector v. The solution vector v is said to be an eigenve
Fourier Transform - from Wolfram MathWorld
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SEARCH MATHWORLD
Algebra
Applied Mathematics
Calculus and Analysis
Discrete Mathematics
Foundations of Mathematics
Geometry
Fourier Transform in the
Calcul
EE 505 B Fall 2011
Final Exam Solutions
1. [35 points]
(a) [10 points] We use the formula
f X (x) =
f U (u)
d
du g(u) u = u i
i
x
There is only one solution u1 that sat1
ises the equation x = ln(u):
x
u1 = e x
ex
Substituting u1 into the pdf:
f U ( u1 ) =
EE 505 B Fall 2011
Final Exam
Nota bene: For full credit you must show all of the steps or reasoning that you used to
get the answer.
1. [35 points] U is a uniformly distributed random variable with probability density
function
1 0u1
f U (u) =
0 for all o
453.701 Linear Systems, S.M. Tan, The University of Auckland
9-1
Chapter 9 The Discrete Fourier transform
9.1
Denition
When computing spectra on a computer it is not possible to carry out the integrals involved in
the continuous time Fourier transform. In
Euler's Identity
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Euler's Identity
Euler's identity (or `theorem' or `formula') is
(Euler's Identity)
To `prove' this, we will first define what we mean by `
, is assumed
EIGENVALUES AND EIGENVECTORS
1. Definition
They are dened in terms of each other. Let A be an n n matrix. A vector
v = 0 is an eigenvector of A with eigenvalue if the equation
Av = v
is satised.
Note that eigenvectors are not uniquely dened:
If v is an ei
EE 500 PMP Colloquium/Seminar - UWEE - EE 500
Academics
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Academics > Course Info > Class Home Pages
Undergraduate
Master's & Ph.D.
Professional Programs
EE 500F
PMP Colloquium/Seminar
Non-Degree Options
Course Info
Autumn 2011
Class Home Pages
Section 17.9 The Divergence Theorem
Turning a Flux Integral into a Triple Integral The last result we consider is a generalization of Greens Theorem converting a ux integral over a closed surface into a triple integral over the interior of the surface. 1.
Divergence Theorem Examples
Gauss' divergence theorem relates triple integrals and surface integrals.
GAUSS' DIVERGENCE THEOREM Let F be a vector field. Let W be a closed surface, and let e be the region inside of W . Then: ( ( F . A oe ( ( ( divaFb .Z
W
DFT Definition
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DFT Definition
The Discrete Fourier Transform (DFT) of a signal
where `
may be defined by
' means `is defined as' or `equals by definition', and
The sampl
Delta Function - from Wolfram MathWorld
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Discrete Mathematics
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Delta Function in the
Calculus and An
1.4 The Matrix Equation Ax b
Linear combinations can be viewed as a matrix-vector
multiplication.
Definition
If A is an m n matrix, with columns a 1 , a 2 , , a n , and if x is in
R n , then the product of A and x, denoted by Ax, is the linear
combination
EE 505 B Fall 2011
Assignment 8 Solutions
For full credit, you must show all of the steps or reasoning that you used to get the
answer.
1. (a) The denition of the region can be written as
B = cfw_( x1 , x2 ) : x1 > 0, x1 < x2 < x1 .
The region is shown i
EE 505 B Fall 2011
Assignment 8
For full credit, you must show all of the steps or reasoning that you used to get the
answer.
1. A joint probability density function is given as
c e x1
x1 > 0 and | x2 | < x1
0
f X1 X2 ( x 1 , x 2 ) =
otherwise
(a) Draw th
EE 505 B Fall 2011
Assignment 7 Solutions
1. (a)
Ecfw_ X =
=
x f X ( x ) dx
b
a
x
dx
ba
b
=
=
=
=
x2
1
ba 2 a
1
( b2 a2 )
2( b a )
(b a)(b + a)
2( b a )
b+2
2
(b) We use the direct method to nd the probability density function for Y:
f Y (y) =
i
f X (x)
EE 505 B Fall 2011
Assignment 7
For full credit, you must show all of the steps or reasoning that you used to get the
answer.
1. Let X be a random variable with a uniform probability density function.
f X (x) =
1
b a
0<a<x<b
0
otherwise
(a) Find Ecfw_ X .
EE 505 C, Autumn, 2010
Vector Calculus
1
1 Calculus Review
1.1 Integration over a plane (double integrals)
Integration over a plane is a special case of integrating over a surface, in which the
surface is a plane. The integral is
f ( x, y) dA
D
where D is