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**Unformatted text preview: **MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Outline #2 (Vector Spaces) Last modified: October 4, 2007 Report errors by email to [email protected] Reading. Luenberger, Section 2.1 - 2.9 Lecture topics. 1 1. Axioms for a Vector Space Here is Luenberger’s notation: • The vector space is X , elements are x,y,z, ··· (no bold or arrows). • The underlying field is R (occasionally C , but that will be made ex- plicit. Elements are α,β, ··· . • The zero vector is θ . • - x means (- 1) x . With these conventions, write down a list of axioms for X . What standard axiom of group theory becomes a theorem with this list of axioms? Prove it. Which axiom on the list can be proved from the others? (Proof left to the homework). 2 2. Independence of the axioms Let X = R 2 . Define addition as usual, but define scalar multiplication by α ( b,c ) = ( αb, 0) Is X a vector space? If not, what axiom is not satisfied? For the NSA budget, let X = R ∪ S . The symbol S denotes a secret amount. For elements in X other than S , addition and multiplication are defined as usual. Here are the rules for S . ∀ x ∈ X, S + x = x + S = S . ∀ α ∈ R ,α S = S . Is X a vector space? If not, what axiom is not satisfied? 3 3. Examples of vector spaces over R . Either say “yes” or specify what axiom is not satisfied. Remember to check for closure. Specify the dimension of the space. (a) All 2 × 3 matrices with real entries. (b) All infinite sequences, e.g. x = (1 , 4 , 9 , 16 , ··· ). (c) All bounded infinite sequences (d) All infinite sequences that converge to zero. (e) All infinite sequences with only finitely many nonzero terms. (f) All infinite sequences for which the terms form a convergent series. (g) All functions f : { A,B,C } → R . (h) All polynomials whose degree is exactly 3. (i) All polynomials of any (finite) degree. (j) All power series. (k) All continuous functions on [0,1]. (l) All linear functions f : R 3 → R Find examples of subspace relationships in the list above. 4 4. Operations on subsets and subspaces Prove that if M and N are subspaces of X , so is M ∩ N . Give an example where M and N are subspaces of X , but M ∪ N is not a subspace. Draw a diagram to illustrate the sum of two disjoint squares, one of side 1 and the other of side 2. The diagram makes is look as though you are adding points, but you are really adding vectors. The tails are all at the origin, and the heads lie in the squares. When the sets that you are adding are subspaces, you get all linear com- binations of a basis for each subspace, so it should be no surprise that the sum is a subspace. Still, as a useful exercise in turning definitions into proofs, let’s prove that if M and N are subspaces of X , so is M + N (p....

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