Properties of bases

RR
by Rose Ann Mae Ramos1 view·Updated Jan 31, 2023
Create your own

Chapters

Audio TranscriptAuto-generated

  1. Good day everyone Welcome to adult 6 10.
  2. My name is Rose and maybe Ramos and I'm here to talk about the properties of pieces.
  3. So at the end of this lesson,
  4. we are going
  5. to determine
  6. the properties of bases and enlightened by the examples of the basis
  7. in a vector space.
  8. First of all, let us define
  9. pieces.
  10. A basis is a set of linearly independent vectors that can be used as a building blocks
  11. to make any other vectors in the vector space.
  12. A space comprised of vectors collectively with
  13. associative and communicative law of addition,
  14. vectors and also the associative and distributive process of
  15. multiplication of vectors by scholars is called vector space.
  16. So a space or a vector space consists of set of V which are
  17. elements of VR called vectors in the field F which are elements of F.
  18. R. Scholars. And the two operating shots, which is the
  19. addition and multiplication
  20. we have here the properties of bases.
  21. So first B is a basis. Second B is a maximal linearly independent set.
  22. Be
  23. next B is a minimal spanning set.
  24. And
  25. next if a B or subsets of V such that A is a linearly independent
  26. and B spans V then
  27. set A is
  28. less turn or equal to the set B.
  29. Next one. All bases for V had the same cardinality.
  30. Hence it is possible to find the dimensions of a vector space
  31. V and W R S. A morphing
  32. if their bases have the same
  33. cardinality.
  34. So as we can see
  35. in the properties of basis property one is actually a
  36. special case of either property to or property three.
  37. So if we take null as the given linearly independent seven V. Or in B.
  38. And apply property two, we obtain property one right.
  39. So likewise, if you take B as the given spanning set of B
  40. and apply property three, we gain
  41. or obtain
  42. the property one.
  43. So they are just interrelated.
  44. So these properties can be generalized to a left
  45. or right vector space over a division ring.
  46. However,
  47. the most of the properties and basis can be
  48. generalized to an arbitrary module over an arbitrary ring.
  49. Say, for example, not all modules have been assessed right?
  50. But we do have the following,
  51. say for example, let em be a left module over a ring. Are then
  52. if Emma's finite visas, then all business for M are finite. Right.
  53. And that
  54. next if amazon infinite visas, then all basis for em have the same cardinality,
  55. people.
  56. So we have to take note that when a module has a basis, then
  57. we can call it a free module.
  58. Other characterizations are possible.
  59. So free module
  60. behave a bit like vector space.
  61. So, however, unlike a vector space, one may be or may not be able to define dimension
  62. on a free module.
  63. It is a set
  64. or it is possible that in a finite
  65. in an infinitely
  66. again,
  67. it is possible that enough.
  68. Finally generated free module,
  69. there are two bases of different cardinality is
  70. So we have here
  71. some examples of
  72. properties and this is
  73. what's up here. So if B is
  74. or if this set be
  75. is linearly independent. Set of vectors in RN or subspace of RN. But does that span R.
  76. N?
  77. Then we can create a basis by adding some vectors to be
  78. So if we add an element double U. To be, then W is linear combination of elements of B.
  79. So since the spanning set, therefore B is a maximal linearly independent.
  80. Say, for example, the set this set
  81. It's linearly independent, but it is not basis of our three
  82. Because it does not span R three.
  83. We can create a basis for R. three by adding
  84. this set to the set B.
  85. Or this
  86. This 1 - 70.
  87. Next one. If this set be spans R. N. But not linearly independent.
  88. Then we can create a basis. Very moving some vectors from be safer.
  89. Why? Because if B is a maximal linearly independent, then any W.
  90. That we add will make it dependent. Therefore
  91. there is an element which is linear combination of its
  92. press adores
  93. I mean
  94. the doors and it has to be W. Hence be spans V.
  95. If you remove a Visa one from B, B no longer responds V.
  96. Since Visa one is not a linear combination of Visa. Once. Right.
  97. Say, for example, the set the set
  98. 100153.
  99. It spans to our end.
  100. It is not a basis because it is not linearly independent set.
  101. We can create a basis for our end by removing the 53
  102. from B
  103. to make it
  104. linearly
  105. independent.
  106. Okay,
  107. so that's the we have here. If B if this set B is a basis for R. N,
  108. then every vector in RN
  109. can be expressed in exactly one way as linear combination of vectors in S.
  110. So
  111. if B is minimal, spanning
  112. B is planning,
  113. if V is dependent, then there is a visible one,
  114. which is a linear combination of Visa once,
  115. but then be without visa one would still be
  116. a spanning set in contradiction to minimal itty.
  117. Say, for example, the set 1001 is a Visas for our end.
  118. Every vector IN RN can be expressed in exactly one way as a linear combination 1001.
  119. So that's it for the properties of business.
  120. It may just be limited. But I hope you learn something.
  121. Thank you so much. And God plans
  122. Again. This is rose and media Thomas from section two
  123. have a great day. Bye!