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Lecture

Lesson 14b -Vector Spaces.pdf

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Department
Mathematics
Course
MATH 1107
Professor
All Professors
Semester
Fall

Description
07102013VectorSpacesWehavetalkedaboutspanbutwhatdoesaspanlooklikeWhatelsecanavectorlooklikeWhatspacedotheyoccupyVectorSpaceAVectorspaceisasetofobjectscalledvectorsthathavethefollowingpropertiesAVectorspaceisasetofobjectscalledvectorsthathavethefollowingpropertiesTherearetwooperationsadditionandmultiplicationthatsatisfy1uvwuvw2uvvu3Thereisazeroelementthathas0vvforeveryvpropertyvv04Everyelementvhasanadditiveinversevthathasthe5AuvAuAvwhereAissomescalar6ABABhAdBl6ABuAuBuwhereAandBarescalars7ABuABuwhereAandBarescalars8Thereisascalar1thatmakes1uuforeveryuInadditionanytimeweperformanoperation18abovetheresultremainsinthevectorspaceExampleofVectorSpaceRealNumbersTherealnumbersmakesavectorspaceOuroperationsareregularandx1uvwuvwWecanaddthefirsttwoorthelasttwobuttheresultwouldbethesame2uvvuTheorderinwhichweaddrealnumberswillnotchangetheresult3Thereisazeroelementthathas0vvforeveryvTherealnumber0worksthisway4Everyelementvhasanadditiveinversevthathasthepropertyvv0Negativenumbersaddtopositivenumberstomake0AuAvwhereAissomescalar5AuvWecanexpandmultiplicationwithrealnumbers6ABuAuBuwhereAandBarescalarsThisisthesameasdistributionwithrealnumbers7ABuABuwhereAandBarescalarsItdoesnotmatterwhatorderwemultiplyrealnumberseveryu8Thereisascalar1thatmakes1uuforThenumberonemultipliedbyanyrealnumberwillreturnthatrealnumber1
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