HW1-250A,B - HOMEWORK 1(Math 250 A B i Let W1 and W2 be...

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Unformatted text preview: HOMEWORK 1 (Math 250 A, B) i. Let W1 and W2 be subspaces of a vector space V. Then: (20 pts) (a) Show that W1 A W2 is a subspace of V. (b) Is Wl U W2 a subspace of V? If your answer is no, provide a counterexample. 2. Show that the set of all symmetric matrices, i.e. W 2 {A e M m (R) | A’ = A} , is a subspace of M m (R). Exhibit a basis for Wandifind the dim(W) . (20 pts) (Hint: Try to find the form that a symmetric matrix A has in general. That will help you find the elements that A can be decomposed into, in other words the elements that span W) 3. Show that the set W = {p(x) e P2[x] l p(x) = p(—x)}. Show that Wisa subspace of P2 [x] . Exhibit a basis for Wand find the dim(W) . (20 pts) (Hint: Try to find the form that a polynomial p(x) in Whas in general. That will help you find the elements that p(x) can be decomposed into, in other words the elements that span W) 4. Let S be a set. For each s e S the function 253 : S —>R defined by: (20 pts) (0 _ l , t = S ZS _ 0 , t i s is called the characteristic function of s . Denote the set of all characteristic functions by X. Consider also the set of all functions f : S ——> R , denoted by F (S). Show the following: (a) IfS is finite, say _S = {s,,sz,...,sn}, then XS} ,1” ,...,;(SH form a basis for F(S). (b) IfS is infinite, then the dim(F(S)) = 00. Could 131,152 ,0”an , serve as a basis for F (S) ? (Hint: For the lst-half of (1)), use aproof by contradiction, considering a maximal independent set and also the fact that 151 , 132 , m, ZS , could be shown to be linearly independent. For the 2nd- half of (b), find a function that cannot be expressed as finite comb’s of 251 , 152 ,..., ZS" , ...) 5. If fl (x), f2 (x),..., f" (x) e C (""1) (IR), i.e. they are (n —l)-tirnes differentiable functions, then the determinant defined by: (20 pts) f1 (x) f2 (x) f.. (x) W) 2 fi’tx) no) no) fi‘"“’(x) f2‘"‘”<x> f.‘"“’(x> is called the Wronskian of f1 (x), f2 (x),t.., f” (x) . This determinant is useful for shoWing us whether the functions fl (x), f2 (x), ..., f" (x) are linearly independent or not. For example, if W(x) ¢ 0 , then f1 (x), f2 (x),..., f" (x) are linearly independent. Show the following: (a) The functions 1 , ex, ez" are linearly independent functions in C 2 (R). (b) Is it true that if W(x) : O , then f1(x), f2 (x), ..., f" (x) are linearly dependent? If your answer is no, provide a counterexample. (Him: For (b), try fl (x) = x3, f2(x) = ’x3‘ ) ...
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