Sol1 - ISyE 7682 FALL 2010 – 1st Homework Problem 0.1 Assume that L is a subspace and A ⊆ R n is an affine manifold Show that(a for any x ∈

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Unformatted text preview: ISyE 7682 FALL 2010 – 1st Homework Problem 0.1 Assume that L is a subspace and A ⊆ R n is an affine manifold. Show that: (a) for any x ∈ ℜ n , the set L + x is an affine manifold; (b) for any x ∈ A , the set A− x is a subspace which does not depend on x . (c) A is a subspace if and only if 0 ∈ A . Solution . a) Let x, ˜ x ∈ L + x and λ ∈ ℜ be given. Then, x − x and ˜ x − x are in L , and since L is a subspace, it follows that (1 − λ )( x − x ) + λ (˜ x − x ) ∈ L , or equivalently (1 − λ ) x + λ ˜ x ∈ L + x . Hence, L + x is an affine manifold. b) Recall that A is an affine manifold if and only if it contains every affine combination of its elements, that is, for every integer k ≥ 1: λ 1 ,...,λ k ∈ R λ 1 + ... + λ k = 1 S 1 ⊆ A ,...,S k ⊆ A = ⇒ λ 1 S 1 + ... + λ k S k ⊆ A . (1) Using this implication, we conclude that A + x 1 − x ⊆ A for every x 1 ∈ A . This is equivalent to A− x ⊆ A− x 1 for every x 1 ∈ A . Interchanging the role of x and x 1 in this last inclusion, we see that the reverse inclusion also holds. Hence, A− x = A− x 1 for every x 1 ∈ A , showing that the set A− x does not depend on the choice of the point x ∈ A . To show that A − x is a subspace, we have to show that α [ A− x ] + β [ A − x ] ⊆ A − x for any scalars α and β . Indeed, using implication (1) several times, we obtain α [ A− x ] + β [ A− x ] = bracketleftbig α A + (1 − α ) x bracketrightbig + bracketleftbig β A + (1 − β ) x bracketrightbig − 2 x ⊂ A + A− 2 x ⊆ A− x . We have thus shown that A− x is a subspace and the result follows. c) If A is a subspace then clearly 0 ∈ A . Now if 0 ∈ A then by (b), it follows that A = A− 0 is a subspace. Problem 0.2 Let A : ℜ n → ℜ m be a given map. Show that the following properties are equivalent: (a) there exists T ∈ ℜ m × n and b ∈ ℜ m such that A ( x ) = Tx + b for every x ∈ ℜ n ; (b) A ( λx + (1 − λ ) y ) = λA ( x ) + (1 − λ ) A ( y ) for every x,y ∈ ℜ n and λ ∈ ℜ ....
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This note was uploaded on 08/24/2011 for the course ISYE 7682 taught by Professor Staff during the Fall '08 term at Georgia Institute of Technology.

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Sol1 - ISyE 7682 FALL 2010 – 1st Homework Problem 0.1 Assume that L is a subspace and A ⊆ R n is an affine manifold Show that(a for any x ∈

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