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Unformatted text preview: ECEN 455: Assignment 4 Problems: 1. (LA: 8.1.2) Let V be a vector space over F . Show that the sum of two inner products on V is an inner product on V . Is the difference of two inner products an inner product? Show that a positive multiple of an inner product is an inner product. Let () a and () b be two inner products on V . Furthermore, for v ,u V , define ( u  v ) = ( u  v ) a + ( u  v ) b . Then, the following properties hold. (a) For any u ,v ,w V , ( u + v  w ) = ( u + v  w ) a + ( u + v  w ) b = ( u  w ) a + ( v  w ) a + ( u  w ) b + ( v  w ) b = ( u  w ) + ( v  w ) . (b) For any v ,w V and s F , ( sv  w ) = ( sv  w ) a + ( sv  w ) b = s ( v  w ) a + s ( v  w ) b = s ( v  w ) . (c) For any v ,w V , ( v  w ) = ( v  w ) a + ( v  w ) b = ( w  v ) a + ( w  v ) b = ( w  v ) . (d) If v negationslash = 0 then ( v  v ) a > and ( v  v ) b > , which implies that ( v  v ) = ( v  v ) a + ( v  v ) b > . That is, the sum of two inner products on V is itself an inner product on V . The difference of two inner products is not necessarily an inner product. Suppose that () a is an inner product on V . Then () a () a = 0 is not an inner product, whereas 2 () a () a = () a is obviously an inner product. A positive multiple of an inner product is also an inner product. Let () be an inner product on V and let c be a positive number. Then, for all u ,v ,w V and s F , we have (a) c ( u + v  w ) = c ( u  w ) + c ( v  w ) (b) c ( sv  w ) = cs ( v  w ) = sc ( v  w ) (c) c ( v  w ) = c ( v  w ) = c ( v  w ) (d) c ( v  v ) > if v negationslash = 0 . 2. (LA: 8.1.9) Let V be a real or complex vector space with an inner product. Show that the quadratic form determined by the inner product satisfies the parallelogram law bardbl + bardbl 2 + bardbl bardbl 2 = 2 bardbl bardbl 2 + 2 bardbl bardbl 2 . The parallelogram law can be shown as follows, bardbl + bardbl 2 + bardbl bardbl 2 = ( +  + ) (  ) = (  ) + (  ) + (  ) + (  ) + (  ) (  ) (  ) + (  ) = 2 (  ) + 2 (  ) = 2 bardbl bardbl 2 + 2 bardbl bardbl 2 ....
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 Spring '08
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