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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- Spring '08
- Vector Space, equivalence classes, Coset, 29 J, V/W, term equivalence relation