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MATH 235
Barry Mc Clinchey

nFridayJanuary 10 Lecture 3Spans and subspaces ofConcepts1 Subspace of a vector space 2 A subset which is closed underand scalar multiplication 33 Lines and planes which are subspaces in4 Prove that the intersection of two subspaces is a subspacen5 Determining if a subset is a subspace of6 Span of a finite set of vectors 7 Span of a finite set of vectors is a subspace n8 Basis of a subspace standard basis of9 Finding a basis by removing redundant vectors n31IntroductionUp to now we have referred to the elements of the setas vectors n not simply as the set For reasons which we will make clear later we will now refer to of all ntuples of real numbers but as a vector spaceFor example rather than say the set 33 we will say the vector space nn is a subspace ofif W 32Definition We say that a subset W of the vector space satisfies two conditions n1 W is a nonempty subset of2 W is closed under scalar multiplication and closed under additionThis means that if u wW thenuW for any scalarand uwWEquivalently W is closed under linear combinations This means that if v and w belong to W then so does vw for any scalarsandNote The word equivalently means that W is closed under scalar multiplication closed under addition if and only if W is closed under linear combinations 321 Observationsn is a subspace of itself since it satisfies the two conditionsThe vector space required to be a subspace n Every subspace W ofmust contain the vector 0 Since W is closed under linear combinations W must contain 0v0 Normally we show that W is nonempty by showing that it contains the 0vector 22 x y0x1 0y1 a subspace of 33 ExampleIs W01Answer
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