Unformatted text preview: Problem 0.5 Show that For any nonempty set S ⊂ ℜ 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 de²nition, 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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- Fall '08