# Ch6 - Solutions to Homework 12 Math 110 Fall 2006 Prob...

• Notes
• 2

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

Solutions to Homework 12. Math 110, Fall 2006. Prob 6.1.1. (a) True, directly by definition. (b) True, this is also required by the definition. (c) False, it is linear in the first component and conjugate linear in the second component. (d) False, there are in fact infinitely many inner products. (e) False, it also holds in infinite- dimensional spaces (see the proof, which does not use finite-dimensionality). (f) False, every rectangular matrix has a conjugate transpose (defined the usual way). (g) False when x is fixed. (h) True when x runs over the whole space. Prob 6.1.3. We have: f, g = 1 0 te t dt = ( t - 1) e t 1 0 = 1 , f 2 = 1 0 t 2 dt = t 3 3 1 0 = 1 3 , so f = 1 3 , g 2 = 1 0 e 2 t dt = e 2 t 2 1 0 = e 2 - 1 2 , so g = e 2 - 1 2 , f + g 2 = 1 0 ( t + e t ) 2 dt = 1 0 ( t 2 + 2 e t + e 2 t ) dt = 3 e 2 + 11 6 , so f + g = 3 e 2 + 11 6 . Since e 2 > 7, we can confirm that the Cauchy-Schwarz inequality holds: | f, g | = 1 e 2 - 1 6 . Likewise, f + g = 3 e 2 + 11 6 1 3 + e 2 - 1 2 = f + g . The intermediate inequality can be checked numerically or by squaring both sides: 1 3 + e 2 - 1 2 + 2 e 2 - 1 6 = 3 e 2 - 1 6 + 2 e 2 - 1 6 3 e 2 - 1 6 + 2 = 3 e 2 + 11 6 .

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

This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern