Vectors space definition with axiom classification

The topic of vector space seems very difficult for the engineering students, in this presentation some of my students tried to explain

1. GOVERNMENT ENGINEERING COLLEGE BHUJ

2. PRESENTED BY :- 150150113011 150150113012 150150113013 150150113014 150150113015 150150113016 150150113017 150150113018 150150113019 150150113020 SUBMITTED TO :- PROF. K.K.POKAR

3. HELLO FRIENDS . I AM BUNNY AND I WILL HELP YOU TO UNDERSTOOD VECTOR SPACES AND SUB SPACES

4. 1. VECTOR SPACES DEFINATION AXIOMS example 2. SUB SPACES DEFINATION PROPERTIES Example INDEX SELECT TOPIC YOU WANT TO ENTER

5. VECTOr SPACES A vector space is a nonempty set V of objects, called vectors on which are defined two operations, called vector addition and scalar multiplication . DIDNT UNDERSTAND ,LETS TRY THIS . CLICK HERE

6. ADDITIVE AXIOMS MULTIPAL AXIOMS BELONGING AXIOMS AXIOMS

7. AXIOM 1 Closure Under Addition For every pair of element u and v in V there corresponds a unique element in V called the sum of u and v denoted by U V +   V Here V is set and U and v are element

8. AXIOM 3 Commutative Law For all u and v , we have U V + = U V +

9. AXIOM 4 Associative Law For all u and v , we have U + V = V + U W W ( ) ( ) + +

10. AXIOM 5 Existence of Zero Element There is an element in V denoted by O such that U U + + = = U

11. AXIOM 6 Existence of Negative For every u in v , the element (-1)u has the property U + = ( -1 ) U

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13. AXIOM 2 Closure Under Multiplication By Real Number (Scalar) For every u in V and every real number k there corresponding's an element in V called the product k and u, denoted by ku .

14. AXIOM 7 Associate Law For Every u in V and all real number k and m, we have K ( ) M U K M ( U ) =

15. AXIOM 8 Distributive Law For Addition in V For all u and v in V and all real k , We have K ( U + V ) = K U K + V

16. AXIOM 9 Distributive Law of Number For all u in V and all real k and m , We have K ( M U ) = + K U U M +

17. AXIOM 10 Existence of Identity For every u in V , We have 1 U = U

18. The belonging axioms are as (1) AXIOM 1 (CLOSURE UNDER ADDITION) (2) AXIOM 2 (CLOSURE UNDER MULTIPLICATION) (3) AXIOM 5 (EXISTUNCE OF ZERO ELEMENT) (4) AXIOM 6 (EXISTENCE OF NEGATIVE )

19. A subspace is a vector space inside a vector space. When we look at various vector spaces, it is often useful to examine their subspaces. SUB SPACES FOR MORE CLICK HERE

20. The subspace S of a vector space V is that S is a sub set of V and that it has the following key characteristics S is closed under scalar multiplication: if λ∈ R , v ∈S, λ v ∈S S is closed under addition: if u , v ∈ S, u + v ∈S. S contains , the zero vector. Any subset with these characteristics is a subspace. PROPERTIES

21. HEY!!!!Guys lets move to example

22. EXAMPLE :- Vector space Let V = be the set X = ( ………….+ Y = ( ………….+ we have to show that is an vector space.  

23. (1) Closure : - Since X , Y R and hence ( ) R Therefore X + Y R  

24. Let X = ( ………….+ Y = ( ………….+ Z = ( ………….+ (X+Y)+Z =(( )+ , )+ +……….+ )+ ) = ( + , + +……….+ + ) = ( ( + ) , ( + ) +……….+ + ) ) = X+(Y+Z)   (2) Associative L.H.S R.H.S

25. (3)Existence of zero element Here the zero vector 0 = (0,0,0,……,0) Then for any X X+O = ( ………….+ + (0,0,0,………,0) = ( …………. = ( ………….+ = X  

26. (4)Existence of negative Let X then (-1)X = -X is the additive inverse of X Since –X = ( - - …………. , - X + (-1) = ( …………. , + (- - …………., - = ( , , , ………….., ) = (0,0,0,………,0 ) = 0 –X + X   I can do that!!!!

27. (5)Commutative Law It follows from the commutative in R, X + Y = Y + X That’s easy

28. (6)Distributive law of number Let k ,m R , Then (k + m)X = ((k + m) , (k + m) ,…………, (k + m) ) = ( k + m , k + m , ………., k + m , ) =( k , k , …., k + ( m , m , …. , m = kX + mX  

29. (7)Distributive Law for vector addition K ( X + Y ) = k( ( ……….+ + ( ………….+ ) =k ( , , ,………….., ) = ( k k …….+ k + ( k … … + = kX + kY   Remember k is scalar quantity where X and Y are vector quantity THE ADDITION OF VECTORS QUANTITY

30. (8)Existence of Identity 1 . X =1 . ( …………. , = ( …………. , = X  

31. (9)Associative law for multiplication of numbers For ever x in and all real numbers k and m (km)X = (km) ( …………., = ( km …………., km =k ( m …………., m =k(m ( …………., ) =k(mX)  

32. EXAMPLE OF SUB SPACE Let U be the set of all vectors of form 2r-s 3r r+s r,s R   Note that 2r-s 3r r+s = 2 3 1 r + -1 1     U = SPAN 2 3 1 -1 1 , U is sub space on   s

33. THANK YOU FOR YOUR COOPERATION .IT WAS PRETTY GOOD TO SHARE KNOWLEDGE WITH YOU.

34. VECTOR = MAGNITUDE + DIRECTION 5 mph East Speed Scalar Velocity = Vector I am sure that you got that

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36. X Y Z

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38. X Y Z Plane y-z Is a subspace OH!! That’s subspace

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