1.7 Max Linear Subsets

1.7 Max Linear Subsets - 1.7. 1.7. Maximal Linearly...

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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 PNU-MATH 1.7. 2. Show that the set of convergent sequences is an infinite-dimensional 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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1.7 Max Linear Subsets - 1.7. 1.7. Maximal Linearly...

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