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problems-1-224-S08

problems-1-224-S08 - (in another math book or on the...

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M ATH 224 P ROBLEM S ET 1 D UE : 29 J ANUARY 2008 When you hand in this problem set, please indicate on the top of the front page how much time it took you to complete. Reading. § 4.1–4.3. Problems from the book: 4.1.2, 4.1.6, 4.1.10, 4.1.12, 4.1.14, 4.1.18. 4.2.1, 4.2.2, 4.2.5. Additional problems: 1. Fill out the background survey and come by my office to return it to me. 2. Suppose that a meter stick is broken at two randomly chosen points. What is the probability that the three segments form the sides of a triangle? 3. Let f ( x ) : [ 0,1 ] R be an increasing function. Can you determine whether or not f is integrable on [ 0,1 ] ? Why or why not? 4. Reread the definitions and properties of abstract vector spaces. Let V be a vector space, and let W V be a subspace. For u,v V , we write u W v if u - v W . Look up the definition of the term equivalence relation
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Unformatted text preview: (in another math book or on the internet). a. Prove that ∼ W is an equivalence relation. We denote the set of equivalence classes by the symbol V/W . b. For v ∈ V , let [ v ] denote its equivalence class in V/W . De±ne an addition rule ˜ + on V/W by [ u ] ˜ +[ v ] = [ x + y ] , where x ∈ [ u ] , y ∈ [ v ] , and + is the usual addition in V . Prove that this addition on equivalence classes is well-de±ned, and that it does not depend on the choices of x and y . c. Come up with a de±nition of scalar multiplication on V/W , and show that it is well-de±ned. d. Prove that V/W is a vector space over R . e. Let V = R 3 and W those vectors x y z ∈ R 3 satisfying x + y + z = . Prove that R 3 /W is one-dimensional....
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