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Historical Introduction to Linear Algebra : Linear Algebra

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                                                              A  historical introduction to Linear Algebra explains how ideas about  solving equations, geometry, and arrays of numbers gradually developed into the modern subject we  now call linear algebra. Historical Introduction to Linear Algebra 1. Early Origins – Ancient Civilizations The roots of linear algebra go back thousands of years. Ancient civilizations such as Egypt and China used methods equivalent to solving systems of linear equations. Around 200 BCE , the Chinese mathematical text The Nine Chapters on the Mathematical Art described procedures for solving simultaneous equations using tables of numbers. These methods resemble what is now called Gaussian elimination . 2. Development of Analytic Geometry (17th Century) A major step occurred during the 1600s...
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External Composition : LInear Algebra

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 External Composition :          An external composition means it is a binary operation performed between two different kinds of elements.   For example :      In the definition of vector space , The scalar multiplication condition is an external composition because  the scalar multiplication condition is                   for a, 𝞊 F and 𝜶 𝞊  V ⇒   a𝜶 𝞊  V     In this condition the product  a𝜶 performed between a scalar  a  and a vector  𝜶                                          *** * The fourth dimensional objects * what is a dimension of an object?   * The Euclidean space   ℛ n * Historical Introduction to Linear Algebra #vector #algebra #vector algebra   #iit #jee #mains #dimension #basis ...
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Vector space ( Definition ) : Linear Algebra

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 [ To see video explanation click on  👉 https://youtu.be/Ojr-skYtVi4 ]                                                                                      Linear Algebra                                     Vector Spaces  Vector Space ( Definition ) :               Let V be a set of vectors and F be a field of scalars. Then V is said to be a vector space  under the field of scalars F if  I)  (V,+) is abelian II) The scalar multiplication condition exists in V       i.e. for  a, 𝞊 F and 𝜶 𝞊  V ⇒   a𝜶 𝞊  V III) together with the following 4 inner conditions exists in V. ...
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Problems & Solutions : Sets : Exercise 1(a) : Intermediate

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 Exercise 1(a) : ( problems & solutions )   1. Which of the following are sets? Justify your answer.       (i) The collection of all the months of a year beginning with the letter J.        Solution :              Suppose A is the collection of  months of a year beginning with the letter J are January, June and               July.                ∴ A= { January, June,July }                 This A  is  well-defined i.e. it is unchanged.                        Hence A is a set.       (ii) The collection of ten most talented writers of India.        Solution :               The collection of ten most talented writers of India may vary from ...
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Roaster form of a set : Sets

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                        In set theory,  there are two methods of representing a set :  1. Roaster or Tabular form  2. Set- Builder form Roaster Form :           In roaster form , all the elements of a set are listed, the elements are being                         separated by commas  and are enclosed within braces { } .  For example  1. The set of all even positive integers lessthan 7 is described in roaster form as{ 2,4,6 }.  2. The set of all natural numbers which divide 42 is { 1,2,3,6,7,14,21,42 }  3. The set of all vowels in the English alphabet is { a,e,i,o,u } etc. Note :   In roaster form , the order in which the elements are listed is immaterial. i.e. in the        example 2, the  set in the example 2 can also be written as { 1,7,2,3,14,21,42,6 }  S...
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Well-defined collection of objects : Sets #well #defined #set

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                                    A well- defined collection of objects in the sense that we can definitely  decide   whether a given  particular object belongs to a given collection  or not.  For example :   * The collection of numbers 2,4,6,8,... is  well-defined because these numbers are formed      by     multiply natural numbers with 2.   * The collection of five most renowed mathematicians of the world is not well-defined ,       because   the criterion for determining a mathematician as most renowed may vary from      person to person.   If you have a question in your mind is that  " Is  " Collection "  a synonym to " Set " " then    answer is " not " .  Now we see the difference between a collection and a set.   " Collec...
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Is aob = (ab) / 3 for a,b 𝞊 Q+ is abelian ? : Group Theory

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  Problem :           Show that the set Q + of all +ve rational numbers forms an abelian group under the composition           defined by aob = (ab) / 3 for a,b 𝞊  Q + Solution :          Suppose Q +   is the set of all +ve rational numbers.               Define the operation ' o ' on  Q +   by  aob = (ab) / 3 for a,b 𝞊  Q + .           Now we are going to prove (  Q +  , o ) is abelian.           Closure property :                 Let a , b 𝞊  Q +       i.e a and b are positive rational numbers.                                            ⇒ ab/3 is also a positive rational number     ...
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