This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 lsthalf 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‘ ) ...
View
Full
Document
This homework help was uploaded on 04/18/2008 for the course MAT 250 taught by Professor Aristidou during the Spring '07 term at DigiPen Institute of Technology.
 Spring '07
 Aristidou

Click to edit the document details