tag:blogger.com,1999:blog-90771508323472307772019-04-22T06:23:49.266-07:00E-Pandu.Come-pandu.com is a media blog that provides statistical guides, mathematics about technology and bloggingE-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.comBlogger56125tag:blogger.com,1999:blog-9077150832347230777.post-61705238272952289262019-04-22T06:21:00.001-07:002019-04-22T06:21:15.168-07:00Linear Independence in vector space
Definition.
The vectors v1,
v2, ..., v3 in the vector v space are called linearly independent
if
c1v1
+ c2v2 + ... +
cnvn = 0
resulting in all scalars c1, c2, ..., c3 must
be equal to 0.
Example
1.
The
vectors and are linearly independent, because if
Then,
c1 + c2 = 0
c1 + 2c2
= 0
And the only solution to
this system is c1 = 0, c2 = 0.
Definition.
Vectors v1, v2,
..., E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-32384073928775465212019-04-22T06:16:00.005-07:002019-04-22T06:16:58.152-07:00Linear Combination in Vector
Definition.
A vector w is called a linear combination and vectors v1, v2, ..., vr if the
vector can be expressed in form
w = k1v1
+ k2v2 + ... + krvr
Where k1,
k2, ..., kr is scalar.
Example
1.
Review the vectors u
= (1, 2, -1) and v = (6, 4, 2)
on R3. Show that w = (9,
2, 7) is a linear combination u and v and w’= (4, -1, 8) is not a linear combination u and v.
Solution:
So that w isE-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-43021465553379618542019-04-22T06:13:00.003-07:002019-04-22T06:13:51.108-07:00Vector Section Space
If given a vector space V, then we may form another vector
space that is a subset of S of V and use operations on V. Because V is a vector
space, summarized operations and scalar multiplication always generate another
vector in V. Share the new system uses the S subset of V as a set of course to
be vector space, the set must be covered under sums and scalar multiplication
operations. That is, E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-70507341697171722452019-04-22T06:07:00.000-07:002019-04-22T06:07:00.801-07:00Definition of Vector Space and its Theorem
Scalar addition and multiplication operations are used in
diverse contexts in mathematics. Regardless of the context, however, these
operations usually fulfill the same rules of arithmetic. So the general theory
of mathematical systems involving scalar addition and multiplication will be
applicable to various fields of mathematics. Mathematical systems with this
form are called Vector Rooms or E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-17147717430776692492019-04-16T20:15:00.001-07:002019-04-16T20:15:53.983-07:00Vector Line and Field Equations in 3rd Dimension
In this article, the
notion of vectors is used to reduce the equation of lines and fields in
space-3, which is very useful in other mathematical fields.
Equation Field in Space-3
A field in space-3 can be determined by giving its inclination and by specifying one of its points. The method that makes it easy to explain the inclination is to assign a vector (called normal) that is perpendicular E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-23759736933189668822019-04-16T19:28:00.002-07:002019-04-16T19:28:30.400-07:00The Cross Product of Vectors in 3-Dimensional Space
the use of a vector in
problems of geometry, physics (mechanics, electromagnetic theory, etc.). Very
helpful, if we have an easy way to determine a vector that is perpendicular to
the two given vectors.
Therefore, in this article
we will discuss the concept of cross product by first defining the
definitions for cross product vector u and v.
Definition:
If u = (u1, u2, u3) and v = (v1, v2, E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-83983110369098118402019-04-16T13:23:00.000-07:002019-04-16T13:23:39.048-07:00Dot Products and Vector Projections
In this article we
will introduce a kind of vector multiplication in space-2 and space-3. The
properties of the multiplication arithmetic will be determined and some of the
applications will be given.
Suppose u and v are two nonzero vectors in space-2 and space-3, and suppose that
these vectors have been located so that the starting point coincides. What we
mean by the angle between u and vE-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-62183127305843678482019-04-16T11:58:00.000-07:002019-04-16T11:58:01.055-07:00Calculating the Length of a Vector
The length of a vector v is often called the v norm and is expressed as |v|. Let v = (v1, v2) be a vector in space 2, then the norm of
vector v is expressed as,
And it is illustrated with Figure 1,
Suppose that v = (v1, v2,
v3) is a vector in space
3. Using Figure 2,
Then we get it
so,
If P1(x1,
y1, z1) and P2(x2, y2, z2)
are two points in space 3, then the distance dE-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-65687069066379966052019-04-16T11:35:00.002-07:002019-04-16T11:35:24.791-07:00Definition of Vector and Scalar Linear Algebra
In this article we will
discuss the meaning of vectors, especially for vectors on R2 and R3.
Many quantities in physics such as force, speed, acceleration, displacement,
and shift are vectors that can be expressed as directional line segments. The
algebraic view, examines the properties of algebra from a vector space, that
is, the properties of vector addition and scalar vector multiplication. E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-4642076012051093262019-04-02T08:38:00.000-07:002019-04-02T08:38:13.544-07:00Application of Inverse Matrices in Linear Equation Systems
In this article we will
produce more about linear equations and invertibility matrices. So we will find
a new method for solving unknown unknowns.
Theorem 1.
If A is a matrix of n x n
which can be reversed, then for each B matrix measuring n x 1, the system
equation AX = B has exactly one solution, namely X = A-1B.
Proof:
Because A (A-1B)
= B, then X = A-1B is a solution AX = B. To E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-88824556593421921022019-04-02T08:14:00.000-07:002019-04-02T08:14:28.208-07:00Elementary Line Operation Methods and Determinants for Finding the Inverse Matrix
In this article E-Pandu
will explain several methods to find inverse matrix values that are larger
than 2 x 2 using elementary and determinant row operations.
Calculating Inverses with Elementary Line Operations (ELO)
A matrix n x n is called
an elementary matrix if the matrix can be obtained from a matrix of units of n
x n namely In by performing a single elementary row operation.
ExampleE-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-83220035624074310602019-03-29T09:57:00.000-07:002019-04-02T08:14:46.015-07:00Definitions of Inverse Matrices and Proof of Inverse Matrix Formulas
In this article, we will
discuss the definition of inverse matrices that solve using elementary row
operations and apply these operations simultaneously to In to get A-1.
Often it will not be known
in advance whether a matrix can be reversed. If the matrix cannot be reversed,
then the matrix is in the form of a reduced line echelon that has at least a
zero number row and will appear on the E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-30021922381323195572019-03-29T09:36:00.000-07:002019-03-29T09:36:52.564-07:00Rules for Calculating Determinants
In the article Definition
Determinants have explained how to calculate the matrix determinant for a
matrix of 2 x 2 and 3 x 3 using the Sarrus method. In this article we will show
how the determinant of a matrix can be calculated using cofactor expansion and
elementary row operations.
Calculating Determinants
Using Cofactor Expansion
Theorem 1
If A is a
quadratic matrix, then the ai
minor E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com1tag:blogger.com,1999:blog-9077150832347230777.post-49487782892957277542019-03-29T08:14:00.000-07:002019-03-29T08:14:32.564-07:00 Properties of Determinants and Linear Equation System Using Cramer's Rules
In this article we will develop some fundamental
characteristics of determinants. The following theorem shows how elementary row
operations on the matrix will affect the determinant value.
Theorem 1.
If A is any matrix squared, then det det (A) = det (At).
Example 1
Theorem 2.
If A and B are quadratic
matrices of the same size, then det (AB) = det (A) det (B).
Example 2.
E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-47063815998281957342019-03-22T23:23:00.001-07:002019-03-22T23:23:26.616-07:00Definition of Determinants
Determinants are functions
with quadratic matrix variables and real values. In this article we begin the
study of real value functions of a matrix variable, which is a function that
associates a real number f (x) with
an X matrix. Assessment of a function
like that is called a determinant function.
The definition of
determinant will be given inductively using minor and cofactor definitionsE-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com1tag:blogger.com,1999:blog-9077150832347230777.post-42527511572176025922019-03-19T09:34:00.002-07:002019-03-19T09:34:37.381-07:00Types of Matrices
Quadratic
Matrix is a matrix whose number of rows is equal to the number of columns.
The zero matrix is a matrix with all entries equal to zero
Matrix properties zero:
A
+ 0 = A (if the size of the matrix A = matrix size 0)
A0
= 0; 0A = 0 (if multiplication requirements are met)
The Diagonal Matrix is a quadratic matrix where all of the entries outside the main E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-12177885182161632232019-03-19T08:54:00.001-07:002019-03-19T09:43:26.848-07:00Addition, subtraction, multiplication, and transpose on the matrix
Addition of Matrices
If A and B are any two
matrices of the same size, then the number of A + B is the matrix obtained by
adding the corresponding entries in the two matrices together. Different size
matrices cannot be added.
Example 1.
because the order size is different, namely A and B measuring 2 x 2 while C is 2 x 3, A + C and B + C are not defined.
If B is any matrix, then E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com3tag:blogger.com,1999:blog-9077150832347230777.post-64034363859462340552019-03-19T08:22:00.003-07:002019-03-19T08:22:46.564-07:00Definition of Matrices
In everyday life a list
often contains numbers arranged in columns and rows, such as food items with
prices and nutrient levels arranged like the following table,
Material
Nutritional
content per kg
Price
Rupiah
per Kg
Protein
Fat
Carbohydrate
Milled Rice
68
7
789
2800
Potato
17
0.85
162.35
3000
Know
E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-50554190762590220862019-03-19T08:05:00.002-07:002019-03-19T08:05:49.974-07:00 Homogeneous Linear Equation System
A
system of linear equations is said to be homogeneous if all constant terms are
zero, that is, the system has the form:
Each system of homogeneous linear equations is a consistent system, because
x1=0,
x2=0, ..., xn=0
always a solution. The solution is called trivial solution. If there is another solution, it is called a nontrivial solution,
Because a system of homogeneous linear E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-31235566198437167072019-03-19T07:46:00.000-07:002019-03-22T22:40:52.550-07:00Gauss Elimination and Gauss-Jordan Elimination
The elimination method for
completing the system of linear equations in the Article Linear Equivalence
System in the Form of Matrices in principle is to change the complete matrix of
the system of linear equations into another simpler matrix with Elementary Line
Operations (OBE). The last form is said to be in the form of a reduced
row-echelon form. A matrix n x m is called the reduced row E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-22858313416288846312019-03-19T07:35:00.003-07:002019-03-22T22:41:22.376-07:00Linear Equation System in the Form of Matrices
An arbitrary system
consisting of ni linear equations with n unknown numbers will be written
as,
The system of linear equations can be written in the form of a matrix,
This line is called an enlarged matrix or an augmented matrix
Example 1
can be represented by:
The basic method for solving a system of linear equations is to replace the system provided with a new E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-36085429336342451912019-03-19T07:01:00.000-07:002019-03-26T07:41:16.893-07:00 Introduction of Linear Equation Systems
An algebraic line in the
xy plane can be expressed by a form equation,
This kind of equation is
called a linear equation in the variables x and y. More generally, we define a
linear equation in n variables x1, x2, ..., xn
as an equation that can be expressed in the form of:
Where a1, a2, ..., an
and b are real numbers.
Example 1,
The following are linear equations:
Note E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-90576216820478161442019-03-14T06:45:00.000-07:002019-03-26T07:46:49.141-07:00Estimated Parameters of the decimated Poisson DistributionThe random X variable is said to follow a truncated Poisson distribution with a parameter λ, if it has the following probabilities function:
The expectations of the random X variable that follows the decimated Poisson distribution above are
Suppose X1, X2, ... Xn is a random sample of size n from a decrepit Poisson distribution, then the estimated parameter λ can be obtained by equatingE-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com3tag:blogger.com,1999:blog-9077150832347230777.post-28659574684553509282019-03-13T09:59:00.000-07:002019-03-13T09:59:48.124-07:00Poisson Random Variable Theory and Poisson Calculation in MATLAB
A random X variable that
takes the values 0, 1, 2, ... is called a Poisson random variable with a
parameter λ if for an λ> 0,
Equation defines a mass function of probability cause
The distribution of Poisson Probability was introduced by S.D. Poisson in a book he wrote about applying the theory of opportunity in the matter of lawsuits, criminal trials, and the like. This bookE-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-19636639094234620042019-03-12T05:38:00.001-07:002019-03-12T05:38:16.111-07:00Combination enumeration
Often we are interested in
determining the number of groups r
different objects that can be formed from a number of n objects. For example, how many groups of 4 different objects can
be taken from 5 objects A, B, C, D, and E ?. To answer this,
reason as follows. Because there are 5 ways to choose the first object, 4 ways
to choose the second object, and 3 ways to select the third object, then E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0