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Unformatted text preview: 1.7. 1.7. Maximal Linearly Independent Subsets 1. Label the following statement as true or false. (a) Every family of sets contains a maximal element. (F) ( ∵ ) Let F be the family of all finite subsets of an infinite set S . then F has no maximal element. (b) Every chain contains a maximal element. (F) ( ∵ ) If A : a partial ordered set and every chain ( 6 = ∅ ) of A has an upper bound, then A has a maximal element. (ex) Z , Q , R (c) If a family of sets has a maximal element, then that maximal element is unique. (F) ( ∵ ) Let S = { a = x 3 2 x 2 5 x 3 , b = 3 x 3 5 x 2 4 x 9 , c = 2 x 3 2 x 2 +12 x 6 } then { a,b } , { a,c } , { b,c } are maximal linearly independent subsets of S So maximal element need not be unique. (d) If a chain of sets has a maximal element, then that maximal element is unique. (T) (e) A basis for a vector space is a maximal linearly independent subset of that vector space. (T) (f) A maximal linearly independent subset of a vector space is a basis for that vector space. (T) 50 PNUMATH 1.7. 2. Show that the set of convergent sequences is an infinitedimensional subspace of the vector space of all sequences of real numbers. ( See Exercise 21 in Section 1.3.) (i) By the exercise 21 of section 1.3,(i) By the exercise 21 of section 1....
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 Spring '09
 RUDYAK
 Linear Algebra, Maximal element

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