4solution - ECEN 455: Assignment 4 Problems: 1. (LA: 8.1.2)...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

Page1 / 6

4solution - ECEN 455: Assignment 4 Problems: 1. (LA: 8.1.2)...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online