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 welldeﬁned, 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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 Spring '08
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 Linear Algebra, Algebra, Vector Space

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