hm1 - Problem 0.5 Show that For any nonempty set S n , a S...

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ISyE 7682 Fall 2010 – 1st Homework Problem 0.1 Assume that L is a subspace and A ⊂ R n is an afne maniFold. Show that: (a) For any x 0 ∈ ℜ n , the set L + x 0 is an afne maniFold; (b) For any x 0 ∈ A , the set A− x 0 is a subspace which does not depend on x 0 . (c) A is a subspace iF and only iF 0 ∈ A . Problem 0.2 Let A : 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 . Problem 0.3 Show that: a) a cone K R n is convex iF, and only iF, x + y K For every x,y K ; b) iF K is a convex cone containing 0, then the set K ( K ) is the largest subspace contained in K ; c) K R n is a subspace iF, and only iF, K is a convex cone containing 0 such that K = K . Problem 0.4 Let A ⊂ R n be an afne maniFold and let X and Y be subsets oF R n such that X ⊂ A . Show that For every λ R , [ λX + (1 λ ) Y ] ∩A = λX + (1 λ )[ Y ∩A ] .
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Unformatted text preview: Problem 0.5 Show that For any nonempty set S n , a S = a (cl S ) = a (co S ) = a ( co S ) , co S = co (cl S ) = co (co S ) . Problem 0.6 Let A R n be an afne maniFold. By denition, a set B A is an afne basis For A iF a( B ) = A and the elements oF B are afnely independent. Use what you already know about linear subspaces to show: (a) A has an afne basis; (b) every afne basis For A has cardinality equal to dim( A ) + 1; (c) B is an afne basis For A iF and only iF B is a minimal element with respect to the property that a( B ) = A ; (d) iF S R n is such that a( S ) = A then S contains an afne basis For A ; (e) every x A can be expressed in a unique way as an afne combination oF the elements oF an afne basis B . 1...
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