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s11_mthsc853_hw01

# s11_mthsc853_hw01 - S = S x 1 ∪ x is a basis for span S(a...

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MthSc 853 (Linear Algebra) Dr. Macauley HW 1 Due Monday, January 24th, 2011 (1) (a) Show that there are no proper subfields of Q . (b) Show that Q ( 2) = { a + b 2 | a, b Q } is a field. (2) Let X be a vector space over a field K . Let 0 be the zero element of K and 0 the zero- element of X . Using only the definitions of a group, a vector space, and a field, carefully prove each of the following: (a) The identity element e of a group is unique. (b) In any group G , the inverse of g G is unique. (c) 0 x = 0 for every x X ; (d) k 0 = 0 for every k K ; (e) For every k K and x X , if kx = 0 , then k = 0 or x = 0 . (3) Let X denote the vector space of polynomials in R [ x ] of degree less than n . Are the vectors x 3 + 2 x + 5, 3 x 2 + 2, 6 x , 6 linearly independent in X ? (Assume that n 4.) (4) The following is called the Replacement Lemma : Let X be a vector space over K , and let S be a linearly independent subset of X . Let x 0 span( S ) with x 0 6 = 0. Prove that there exists x 1 S such that the set
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Unformatted text preview: S = ( S \ { x 1 } ) ∪ { x } is a basis for span( S ). (a) Prove the Replacement Lemma (b) Suppose that B is a basis for X containing n elements, and let B be another basis for X . Show that | B | = n . (5) Let Y be a subspace of X , and denote the congruence class mod Y containing x ∈ X by { x } . We can make X/Y into a vector space by deﬁning addition and scalar multiplication as follows: { x } + { z } = { x + z } , { ax } = a { x } . Show that these operations are well-deﬁned, that is, they do not depend on the choice of congruence class representatives. (6) Let X 1 and X 2 be vector spaces over a ﬁeld K . Show that dim( X 1 ⊕ X 2 ) = dim X 1 + dim X 2 . (7) Let Y be a subspace of a vector space X . Show that Y ⊕ X/Y is isomorphic to X ....
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