Linear Algebra may be defined as the form of algebra in which there is a study of different kinds of solutions which are related to linear equations. In order to explain the Linear Algebra, it is important to explain that the title consists of two different terms. The very first term which is important to be considered in the same, is Linear. Linear may be defined as something which is straight. Linear equations can be used for the calculation of the equation in a xy plane where the straight lines has been defined. In addition to this, linear equations can be used to define something which is straight in a three dimensional perspective. Another view of linear equations may be defined as flatness which recognizes the set of points which can be used for giving the description related to the equations which are in a very simple forms. These are the equations which involves the addition and multiplication.
This presentation will be very helpful to learn about system of linear equations, and solving the system.It includes common terms related with the lesson and using of Cramer's rule.
Please download the PPT first and then navigate through slide with mouse clicks.
This presentation will be very helpful to learn about system of linear equations, and solving the system.It includes common terms related with the lesson and using of Cramer's rule.
Please download the PPT first and then navigate through slide with mouse clicks.
A NEW STUDY OF TRAPEZOIDAL, SIMPSON’S1/3 AND SIMPSON’S 3/8 RULES OF NUMERICAL...mathsjournal
The main goal of this research is to give the complete conception about numerical integration including Newton-Cotes formulas and aimed at comparing the rate of performance or the rate of accuracy of Trapezoidal, Simpson’s 1/3, and Simpson’s 3/8. To verify the accuracy, we compare each rules demonstrating the smallest error values among them. The software package MATLAB R2013a is applied to determine the best method, as well as the results, are compared. It includes graphical comparisons mentioning these methods graphically. After all, it is then emphasized that the among methods considered, Simpson’s 1/3 is more effective and accurate when the condition of the subdivision is only even for solving a definite integral.
A NEW STUDY OF TRAPEZOIDAL, SIMPSON’S1/3 AND SIMPSON’S 3/8 RULES OF NUMERICAL...mathsjournal
The main goal of this research is to give the complete conception about numerical integration including Newton-Cotes formulas and aimed at comparing the rate of performance or the rate of accuracy of Trapezoidal, Simpson’s 1/3, and Simpson’s 3/8. To verify the accuracy, we compare each rules demonstrating the smallest error values among them. The software package MATLAB R2013a is applied to determine the best method, as well as the results, are compared. It includes graphical comparisons mentioning these methods graphically. After all, it is then emphasized that the among methods considered, Simpson’s 1/3 is more effective and accurate when the condition of the subdivision is only even for solving a definite integral.
1 Part 2 Systems of Equations Which Do Not Have A Uni.docxeugeniadean34240
1
Part 2: Systems of Equations Which Do Not Have A Unique
Solution
On the previous pages we learned how to solve systems of equations using Gaussian
elimination. In each of the examples and exercises of part 1(except for exercise 1 parts d and e)
the systems of equations had a unique solution. That is, a single value for each of the variables.
In example 3 we found the solution to be 7 23 3, . This means that the graphs of the two lines in
example 3 intersect at this unique point. In 2-space, the xy-plane, we have the geometric bonus
of being able to draw a picture of the solutions to a system of two equations two unknowns.
Clearly, if we were asked to draw the graphs of two lines in the xy-plane we have 3 basic
choices/cases:
1. Draw the two lines so they intersect. This point of intersection can only happen once for
a given pair of lines. That is, the two lines intersect in a unique point. There is a unique
common solution to the system of equations. Discussed in part 1.
2. Draw the two lines so that one is on "top of" the other. In this case there are an infinite
number of common points, an infinite number of solutions to the given system. Discussed
in part 2.
3. Draw two parallel lines. In this case there are no points common to both lines. There is
no solution to the system of equations that describe the lines. Discussed in part 2.
The 3 cases above apply to any system of equations.
Theorem 1. For any system of m equations with n unknowns (m < n) one of the following cases
applies:
1. There is a unique solution to the system.
2. There is an infinite number of solutions to the system.
3. There are no solutions to the system.
Again, in this section of the notes we will illustrate cases 2 and 3. To solve systems of
equations where these cases apply we use the matrix procedure developed previously.
Example 6. Solve the system
x + 2y = 1
2x + 4y = 2
2
It is probably already clear to the reader that the second equation is really the first in
disguise. (Simply divide both sides of the second equation by 2 to obtain the first). So if we
were to draw the graph of both we would obtain the same line, hence have an infinite number of
points common to both lines, an infinite number of solutions. However it would be helpful in
solving other systems where the solutions may not be so apparent to do the problem
algebraically, using matrices. The matrix of the system with its simplification follows. Recall,
we try to express the matrix
1 2 1
2 4 2
in the form 1
2
1 0
0 1
b
b
from which we can read off the
solution. However after one step we note that
1 2 1
2 4 2
1 22 R R
1 2 1
0 0 0
. It should be clear to the reader that no matter what further
elementary row operations we perform on the matrix
1 2 1
0 0 0
we cannot change it to the form
we hoped for, namel.
Increased human activities in the Arctic has led to the diminishment of Arctic sea ice, about 70,000 km2 per year and has raised concerns for the region’s future. Measurements show that the ice has grown thinner, approximately 40% in the last two decades. The region is opened to increased human activities like commercial shipping, Arctic oil and gas exploration, in addition to deposition of soot by the maritime vessels. Black carbon from incomplete combustion is lodging over the ice and is causing graying of ice caps which was once a reflective surface to absorb more of sunlight and warm the water. Increased water temperatures are having grave impacts on the flora and fauna that are dependent on ice. In near future Polar bears are likely to face extinction as their breeding habitat is given to melting ice. Trapped green house gases like methane are released due to the melting areas of permafrost. Some simple maths can give us the glimpse of the complexity faced by the scientists in handling ice-ocean-climate models.
Social construction of race and gender, patriarchy and prejudice and discrimi...Service_supportAssignment
Social construct may be defined as the social mechanism or a category which has been created by the society. It may either be a perception which is created by an individual, a group or an idea which is constructed because of a culture. The present society has created a large number of constructs which are not good. In this research paper, the discussion will be done on the social construction of race and gender and the problems associated with the same. In addition to this, how can social construct forms to be the basis for discrimination and prejudice? Further, racism and sexism will be discussed with examples and the role of power in the same. To end, patriarchy will be discussed and its role in racism and sexism will be added
A daydream has to be sweet and hence the term dream. A coffee day dream for a girl combines not only the right coffee but also gives her a lucid daydream on similar sweet elements, making her a princess in a castle waiting to be rescued by a prince. This pretty much forms the thematic background for the animation advertisement ‘Coffee Daydream’. Visual effects set the mood here as the viewer watches a girl having coffee, she sprinkles sugar on the coffee and when the advertisement is over she has completed her coffee and her dream. The use of visual technology adds the elements for the dream which is very subtly weaved into the story plot. Techniques of compositing, particle effects and smoke animation are used here to represent the goodness of the coffee being advertised for.
Organizational Management has been defined as the style of managing business of an organization is big or small. This management process involves the process of organizing, planning, leading and controlling the resources along with the main essence of achieving the goal of the business as well. The reason why organizational management is approached is that it provides better decision making capabilities which is both effective and useful to the way in which an organization can run and also carry on proper management strategies (Nikolakopulos, n.d.).
Nike has been the most successful organizations which has excelled in the development of the sports good. Over the years it has been consistently maintained its brand and is regarded as the most innovative organization. Over the years, Nike has been producing new and advanced sports accessories and has consistently been the top choice of the sportspersons.
The Coca-Cola Company, incorporated on September 5th, 1919 is a well known beverage company. The company has a ownership and licensing of brands and markets over 500 non-alcoholic beverages brands which usually consist of sparkling beverages but also a varied number of still beverages like water , enhanced water , juice and juice drinks , ready to drink teas and coffees and also many sports and energy drinks. For every industry who has to spend reasonable time and effort in marketing, Coca Cola serves as a true inspiration (World of Coca-Cola, 2015).When most of the company belongs to mature stage of the product life cycle, and is operating in a competitive market with little product differentiation, the company has been successful to grow in terms of strength as a brand and its marketing techniques. (Cokecce.com, 2015)
The recent downturn in the economy and recent failures in the business have been merged for the creation of a financial environment to make reports unlike any other within present memory. There has been a major impact on the confidence of investors that had been shaken up by an increase in the volatility within the markets of capital. This has further been followed up by unsettlement in the extremely publicized restatements being drafted for the statements of finance (Bond and Cummins 2010). These have resulted in the generation of several questions regarding the quality being presented in the reports of finance.
Background: Samsung Electronics Ltd is a multinational electronics company which has many manufactory and distribution centers all around the world. A subsidiary of the Samsung Group that is based out of South Korea, the company at presents is much diversified being in the production and sales of consumer electronics.
The country selected to be analyzed from the perspective of international business is Ireland. This country notebook will have its basis on making an organization to expand its operations to the business world of Ireland which has been regarded due to several reasons as the best country for doing business (Hill 2014). The target market selected is the hospitality industry of Ireland and the organization selected is Double Tree Hilton expansion internationally into the realms of Ireland.
The choice of consumers differs from one product to another. For example in case of skincare products, the choices are much different because youth under the age of 20-25 have different mindsets for the products because these age groups youth look for costly, from reputed brands. The lifestyle has changed as so the demand has changed. The consumer behavior and thoughts are changes as the time passes. The skincare or say as cosmetics products are highly concerns for female than male, and therefore this research also focuses on female participants a most to understand their choice criterion for skin care products and the value they perceive. The aim of this report is to understand consumer prospective, thoughts and choices while buying the skincare or cosmetics products. The results are obtained using SPSS software as quantitative analysis and conclusion is drawn based on that only. The results show that lifestyle; variety of products, and interest level of skin care products and more has impacted the choices of the consumers under the age of 20-25 in China. The analysis suggests that there have been done enough spending in last 5 years in China which is many folds of the last decades spending for skin care or cosmetics products. The value of choice of skin care or cosmetics products are also discussed and analyzed in the reports, through understanding of lifestyle, the kind of products, income and more factors that impacts the value of skin care or cosmetics products. The obtained results from the survey it is observed that most of the youth have various choices for skin care products depending on their budget, lifestyle, education and occupation. The upper age youth who used the skin care products mostly belong to employment as they need these products highly. The other group who used mostly was prescribed by the doctors. The rest were dependent on the income group and interests to have different choices for the skin care products and brand
Title: Millennium Bridge at London - Steel Structure Failure
This is 320 m span aluminum and steel bridge across the river Thames. This bridge has steel structure failure because it had vertical, lateral and torsional stiffness. The problem occurred because of side vibration of the bridge deck because of pedestrian lateral excitation. The main reasons for this failure were lateral stiffness of the deck and low damping potential which happens in steel structure only. This bridge was made of two dimensional cable truss. The stiffness in this bridge structure caused this failure in this bridge. Therefore, this bridge was closed for few days to fix the problem. It is therefore highly relates to steel structure failure and is suitable for the case study as well. This problem was rectified with help of installation of lateral dampers. All these characteristics of this bridge failures relates to steel structure failure.
With time the whole of the world is being influenced by the social media like YouTube (Bennett & Strange, 2011). The impact of YouTube can be seen in some recent TV programs and feature films. Some of the new TV programs and feature films have adopted the key elements of YouTube aesthetics, including the found dotage device and hand – held camera work to influence the viewers of the authenticity of a viewpoint of character or of narrative (Snickars & Vonderau,2009).
The following paper compares and contrasts the three recent feature films and TV programs, including 127 Hours, Paranormal Activity 3, and Blue Valentine. Each of these movies utilized the same element of YouTube aesthetics that is hand held camera work
Sports, Business, Theatre or Drama; change seldom discriminates. It resonates in each and every walk of life. On the brink of a terrific change is Politics, courtesy the social media. Social media has rapidly grown as a forum for political discourse and activism. Its various platforms such as Facebook, Twitter, Instagram, Youtube etc. are providing a plethora of new ways to engage citizens in politics (Benkler, 2006). A great advantage inherent in social media is the possibility of personal, ie., one to one communication. Politicians as well as political parties are seemingly benefitting with this new found ability to reach out to their potential voters. It has become possible for politicians to reach voters in a well targeted manner without relying on the media as an intermediary (Gentle, 2012). Various reactions, messages, feedbacks and debates are generated online. In addition to this, support for offline causes of a political party are also generated through social media petitions
This research proposal is on the effect of the lending of bank capital and the link between the actual financial condition and the real activities that has been going around. This has succeeded in gaining a lot of attention in past few times because of the financial crisis the world has seen. The techniques of panel-regression can be used to study the lending techniques of bank’s large holdings and companies and the effects small or big of capital on lending (Pelosky, 1991). Then the effect of the capital ratios will be concluded using a variant model, and again the researcher will look for the results that are in marked contrast to estimate obtained by using simple practical relations between the aggregate commercial-bank assets and leverage growth, this has recently been very powerful which was kind of influential for policy maker’s as a result point of views regarding how the loan growth is affected by the bank capital. The models which have been estimated will be used to understand the recent developments in bank lending
The expression and gesture of an artist worth more than action, in today’s culture Artistic expression pays a civilly prevalent. In today’s society dance plays a very important role, to communicate their ideas through various medium. The variety form of dance use the oldest expressions from the culture, and experiment it in a new style. For many in the society dance are entertainment, education, stress release, and a form of worship. Beyond everything in life the thoughts of dance and art serves a greater purpose, artist believes that dance denotes the meaning of tradition and the results of its expression depicts the reaction of culture on society. Dance is an initiation of culture and tradition to the human torso in its most raw form, whether one is bringing up their hand or performing in ballet, in whatever way the artist is holding out and holding back the tradition and style alive.
Understanding the social gifts of drinking rituals an alternative framework f...Service_supportAssignment
The drinking behavior of the binge was described as the most significant reason for the health issues in college campus. By interrelating the ritual behavior and the health condition of the students, the authors conducted focus groups discussion. Through the in-depth interview they explored the nature of alcohol consumption was high among college students. This report extracted from the discussion provides a clear picture about the role of Ritual with student drinking in the campus. With the interpretation of the subject “Drinking-as-ritual” in a theoretical Framework let the authors to discuss how developers of public service announcement captured and contextually drinking rituals. This study makes PSA a more relevant to the target audience
Music to me is an art that stands amongst a few other things I enjoy profoundly. Music is something I can turn to in whichever mood I’m in. It can cheer me up when I’m feeling down. It can calm me down when I’m worked up and very often it has been able to give me good energy. It can motivate me when I feel demotivated with anything. Music has also been a great source of inspiration to me. Music alone has not captured me but lyrical content has also contributed towards my love for music
Modern HRM practices try to recruit employees within the organization. The reasons for internal recruitment are as follows. It is less time consuming in order to hire resources within the company. There is no need for extensive skill assessment process. Employee is well known to the company. There is no need to scout for new candidates. This increases procedural efficiency (Moser, 2005). This process of internal hiring ensures that the employees within the organization sense job security (Chan, 1996). Apart from this it has been observed that new hires joining rate is lesser when compared to internal employees. It takes time for the external candidate to understand the official process of the company. During certain times for some positions there is high employment rate. There is lack of external resources during this time internal sourcing is preferred by the companies (Moser, 2005). This process is saves time. This is found to boost productivity and overall morale of the company
Nike is basically an American establishment for athletic merchandises. It has managed to garner immense brand recognition around the world. They design, manufacture, produce and have an effective distribution system. The company is involved in a number in selling number of sporting gears and athletic accessories (Osayawe Ehigie, 2006). Nike has adopted multi level marketing channels in order to sell. Logistic management of the company is well known. There are a number of factors that needs to be changed in the marketing strategy in order to ensure that Nike presence increases sales in the Nigerian markets and the company manages to expand its consumer base. There is potential for the company emerges as a strong brand in under serviced Nigeria and in African continent. The purpose of this thesis is to look into current Nike presence in Nigeria and recommendations will be proposed based on the analysis
The basics needs of human existences are food, clothing’s & shelter. From times immemorial man has been making efforts in improving their standard of living. The point of his efforts has been to provide an economic and efficient shelter. The possession of shelter besides being a basic, used, gives a feeling of security, responsibility and shown the social status of man.
Every human being has an inherent liking for a peaceful environment needed for his pleasant living, this object is achieved by having a place of living situated at the safe and convenient location, such a place for comfortable and pleasant living requires considered and kept in view.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
1. Linear Algebra and its use in finance:
Introduction
Linear Algebra may be defined as the form of algebra in which there is a study of different kinds
of solutions which are related to linear equations. In order to explain the Linear Algebra, it is
important to explain that the title consists of two different terms. The very first term which is
important to be considered in the same, is Linear. Linear may be defined as something which is
straight. Linear equations can be used for the calculation of the equation in a xy plane where the
straight lines has been defined. In addition to this, linear equations can be used to define
something which is straight in a three dimensional perspective. Another view of linear equations
may be defined as flatness which recognizes the set of points which can be used for giving the
description related to the equations which are in a very simple forms. These are the equations
which involves the addition and multiplication.
The process of learning of linear algebra is tough and complicated. But most of the part of
algebra is concentration towards the objects which are either straight or flat. In doing so, there is
not a requirement to visualize the same.
In this paper, the key concepts which are related to linear algebra which includes the use of
equations, matrices and vectors will be discussed, the theorems related to the same will be
analyzed. After this, the important of Linear algebra will be understood in finance. This will help
to develop a sense of information towards linear algebra.
Application of Linear Algebra
2. Linear algebra forms to be one of the most important part of mathematics. It is a branch which
has a large number of applications in the real world. Because of its ease in solving, it can be used
in different parts of science which the approximation of equations is done with the help of linear
equations. Linear algebra exists in different forms such as abstract algebra, vector spaces and
matrics and have been used for different kinds of theories of mathematics such as the group
theory, ring theory etc (Noble, Ben, 1988). By the process of understanding of different kind of
tools and theorems which are related to linear algebra, it is important to realize the importance of
the functional implementation of the same. One example where numerical application has been
widely used is the Numerical processing of images. This is the branch of algebra where, the
algorithm helps to select the usual information for the discretion of the users. They are also
useful in the linear control theory. The system can be directly described as the part of the state
which is the vector and the change of state is identified by the matrices.
Solutions to linear equations
In order to solve the different kinds of linear equation one needs to understand the forms of
linear equations which has been existing. Suppose, one assumes the following sets of equations:
x2
+ y2
= 1
-x + √3y = 0
In order to solve the equations, the substitution has been done in such a manner that the possible
values of x = √3/2, y=1/2 and x=-√3/2 and y=-1/2. After this one needs to find the plot of the
solutions in the xy plane. In order to do this, a circle is taken whose center falls to be the origin
with a radius of 1 and draw the straight line through the same with the slope 1/√3 in order to get
the equations true(Beezer & Robert Arnold, 2008). After this has been obtained, the desired
solutions will be obtained. The solutions of linear equations can be written assets:
3. S: { (√3/2, ½), (-√3/2, -1/2)}
Definition
A system of linear equation may be defined as the collectionof different kinds of equation in
which the variable is the quantities x1, x2,x3, x4 …xn of the form:
a11x1 + a12x2 + a13x3 + ··· + a1nxn = b1
a21x1 + a22x2 + a23x3 + ··· + a2nxn = b2
a31x1 + a32x2 + a33x3 + ··· + a3nxn = b3
am1x1 + am2x2 + am3x3 + ··· + amnxn = bm
A solution of the system of linear equations may be defined as the set which consists of each of
the solution in the equation and nothing else.
For example:
X1+ 2x2 + x4 = 7
x1 + x2 + x3 -x4 = 3
3x1 + x2 + 5x3 - 7x4 = 1
In the above set of equations, there are n=4 variables and m=3 variables.
The possibilities of the set of solutions in a linear equations can be explained with the following
examples:
2x1 + 3x2 = 3
x1- x2 = 4
While plotting the solutions to the same on an x-y plane one will be getting two lines. While one
of them will be having a positive slope, other will have a negative slope. But they will be having
a common point which is (x1, x2=(-3, 1) which will also be the solution to the equation. Thus,
4. according to the geometry it can be believed that it will be the only possible solution, and it is
unique.
2x1 + 3x2 = 3
4x1+6 x2 = 6
If the plot of these equations is draws, one line will be on top of another. Thus, there will be
many points which will make the equation true.
Another examples can change the possible sets to parallel lines:
2x1 + 3x2 = 3
4x1+6 x2 = 10
This equation will be giving the identical slope or the parallel lines.
Thus, the set of linear equations can possess different kinds of behaviors. It can either be parallel,
meet at a single point or meet at infinite points.
Equivalent system of linear equations:
The two linear equations are equivalent if their solution sets are equal. Let us assume that there is
a set of two linear equations. The following operations will transform them in a different one.
This operation will be called as equation operation.
Theorem 1: Equation operations preserves the solution sets:
According to this theorem, if one of the operations is applied to the system of linear equations,
then the original system and the transformed system can be equivalent.
a11x1 + a12x2 + a13x3 + ··· + a1nxn = b1
a21x1 + a22x2 + a23x3 + ··· + a2nxn = b2
a31x1 + a32x2 + a33x3 + ··· + a3nxn = b3
5. am1x1 + am2x2 + am3x3 + ··· + amnxn = bm
It is assumed that S denotes the solution to the statement of the theorem and T denotes the
solution to the system.
a) S is a subet of T. If (x1, x2, x3….xn)= β1, β, β3… βn € S is a solution for the original
system of equation. If we ignore the i-th equation, it can be said that the other equation of
the transformed system are true:
aαi1 x1+ a α i2 x2 a α i3 x3 …..a α in xn = α βn
This states that the i-th equation of the transformed system is also true, thus β is also a subset of
the equation.
Theorem 2: Three equations one solution, can be solved by done by the following sequence of
equations:
x1 + 2x 2 +2x3 = 4
x1 + 3x 2 +3x3 = 5
2x1 + 6x 2 +5x3 = 6
Now in the above equations, a=-1 times equation is done to get the following solution:
x1 + 2x 2 +2x3 = 4
0x1 + 1x 2 +1x3 = 1
2x1 + 6x 2 +5x3 = 6
Now a=-2 times the equation 2, added to the equation to get the result:
6. x1 + 2x 2 +2x1 = 4
0x1 + 1x 2 +1x3 = 1
0x1 + 2x 2 +1x3 = -2
From the above results, it can be obtained that
x1 + 2x 2 +2x3 = 4
1x 2 +1x3 = 1
x3=4
Theorem: In order to denote the columns of the m*n matrix, A having the vectors as A1, A2, A3-
--An. Thus, x may be defined as the solution to the linear system of equations LS (A, b) if and
only if b is equal to the combinations of the columns related to A which are formed with the
entries of x1.
[x]1 1 A1 + [x] 2 A2 + [x] 3 A3 + ··· + [x]n An = b
Suppose, if there exist an entry into the coefficient matrix A which is related to the row I and the
columns j which has two names aij can be defined as the coefficient of xj.
Matrix and Vector System of Equations:
A m*n matrix may be defined as the rectangular layout of numbers from C having m rows and n
columns. The rows of the matrix can be reference to start at the top and to work down (i.e. row1
is at the top) and the columns will be referenced to start from the left (column 1 is at the left).
The matrix is noted as Ai and the notation is written in the form of: [A]ij.An example of the
matrix with m=3 rows and n=4 columns can be defined as :
-1 2 5 3
1 0 -6 1
-4 2 2 -2
Augmented Matrix:
7. In order to explain the augmented matrix, it is important to consider the system which has m
equations present in n variables. Each of these have the coefficient matrix A and a vector
constraint as b. In such a case the augmented matrix of the system is the matrix in which
m*(n+1) matrix is the one whose first n columns are the one who is the part of Matrix A and the
last column of the same is the column vector b.
The augmented matrix can be written as: [A | b].
Row operations:
Row operation can be performed by the following. This means the swapping of two of the
locations, multiply each entry related to the single row with a non-zero quantity and multiplying
the values to the entries which are there in the same column in the second row.
Theorem:
Row equivalent matrices
Vector
Vectors can be of different types and can include column vectors ordered list consisting of m
numbers, which are written in a vertican order. This starts from the top and proceeds to the
bottom. The column vector can also be referenced as a vectors. The column vectors are noted by
writing the same in bold using smaller alphabets.
8. Definition of Zero column vector:
A zero column vector may be defined as the vector of size m, where each and every entry of the
vector is 0.
0
0
0
Vector of constraints:
For a given system of equations,
a11x1 + a12x2 + a13x3 + ··· + a1nxn = b1
a21x1 + a22x2 + a23x3 + ··· + a2nxn = b2
a31x1 + a32x2 + a33x3 + ··· + a3nxn = b3
am1x1 + am2x2 + am3x3 + ··· + amnxn = bm
vector of constants may be defined as the column vector of size m
b1
b= b2
bn
9. Solution Vector:
Solution vector may be defined as the column vector of size n :
x1
x2
xn
Theorem of Vector Space Properties of column vectors:
It is assumed that the Cm
may be defined as the set of column vectors having the size m with
addition and multiplication scales, then:
• ACC Additive Closure, Column Vectors
if u, v € Cm
, then u + v € Cm
• SCC Scalar Closure, Column Vectors
If α€ C and u€ Cm
then αu € Cm
• CC Commutativity, Column Vectors
If u, v € Cm
then u+v=v+u
• AAC Additive Associativity, Column Vectors
If u, v, w € Cm
, then u+(v+w)= (u+v)+w.
• ZC Zero Vector, Column Vectors
There is also a vector 0 which is known as the 0 vector, such as the u+0=u, for all u Cm
• DSAC Distributivity across Scalar Addition, Column Vectors
If α, β € C and u € Cm
, (α+ β)u= αu + βu
• OC One, Column Vectors
If u € Cm
, u=v.
• AIC Additive Inverses, Column Vectors
If u € Cm
, then there is an existence of a vector where –u € Cm
, so that u+ (-u)=0.
Proof: Some of the properties are the one are obvious, bt it is important that the proof can be
given for them. Though prove for the same is quite tough.
10. For DSAC, the proof can be given as:
[(α+ β)u]i= (α+ β) [u] i
= [αu] ] i + [βu] i
Because of some of the components of the vectors (α+ β) are equal and αu + bu are equal for all,
1<i<m, proves that the vectors are equal.
Theorem Vector Form of Solutions to the Linear equations:
Let us assume that [A | b], may be defined as the augmented matrix which consists of the system
of linear system which is consistent LS(A, b) of m different kinds of equations which are present
in n different variables. Let us assume that B is a row-equivalent of m* (n+1) matrix and it is
reduced in the row-echelon form. Also, if B has r non zero rows, columns without leading 1’s
with the indices and the columns which are leading 1’s having indices D= {d1, d2, d3, ….dr). The
vectors can be defined as:
[c]ij= 0 if i€F
[B] k, fj , if i€D, i=dk
[Uj]I = 1 if i € F, i=fj
0 if i € F, I != fj
-[B]k, fi if i€ D, i=dk
Use of Linear equation in finance:
Linear equation are used in a large number of financial equations and can be used to study a
large number of financial context. Linear algebra can be used for the purpose of representing the
potential schemes related to pricing and can help to compare different models at different points
or price points. There are a number of financial situations where linear equations sort to be very
useful. With the help of the system of calculation of one variable, the other variables can be
easily known.
11. It is used in finance, for a large number of purposes such as to give the amount of interest which
may be accrued on a line of credit after given point of time. In order to explain the example of
the use of linear equation in financial context, a situation can be considered where one needs to
have money to make improvements in the credit line in a given point of time (Berry et al, 1995.
By consideration of the situation and doing the calculations using the linear equations, one can
assume the amount which has to be paid with an interest rate of one year. One can decide to pay
off in a go or through the easy monthly installments as per his choice. In each and every process
of banking, linear equations can be used for the purpose of getting the rate of interest, the total
amount and the number of years which would be required to pay the amount. The complete
banking process is done with the help of linear equations can be implemented with the help of
linear equations.
References
Noble, Ben, and James W. Daniel. Applied linear algebra. Vol. 3. New Jersey: Prentice-Hall, 1988.
Beezer, Robert Arnold. A first course in linear algebra. Beezer, 2008.
Berry, Michael W., Susan T. Dumais, and Gavin W. O'Brien. "Using linear algebra for intelligent
information retrieval." SIAM review 37.4 (1995): 573-595.