My Math Forum  

Go Back   My Math Forum > College Math Forum > Linear Algebra

Linear Algebra Linear Algebra Math Forum

LinkBack Thread Tools Display Modes
April 19th, 2017, 09:22 AM   #1
Joined: Jun 2012
From: Uzbekistan

Posts: 59
Thanks: 0

Inequality with max for independent vectors

Hello everybody.
I have a problem which i can't prove or give a counterexample.

Let $\displaystyle x=\{x_n\}_{n=1}^{\infty}$ and $\displaystyle y=\{x_n\}_{n=1}^{\infty}$ be two linearly independent vectors in infinite dimensional linear space i.e. $\displaystyle \{x_n\}_{n=1}^{\infty}\neq \{\lambda y_n\}_{n=1}^{\infty}$ with following properties:

1) $\displaystyle \max\limits_{1\leq n} \{|x_n|\} = \max\limits_{1\leq n} \{|y_n|\} = \max\limits_{1\leq n} \{|a_{11}x_n+a_{12}y_n|\}
\max\limits_{1\leq n} \{|a_{21}x_n+a_{22}y_n|\}=1$

2) $\displaystyle \max\limits_{1\leq n} \{|x_n+y_n|\} =\max\limits_{1\leq n} \{|(a_{11}+a_{21})x_n+(a_{12}+a_{22})y_n|\}$

3)$\displaystyle \max\limits_{1\leq n} \{|x_n-y_n|\}=\max\limits_{1\leq n} \{|(a_{11}-a_{21})x_n+(a_{12}-a_{22})y_n|\}$.

Where $\displaystyle a_{ij}$ real numbers such that $\displaystyle a_{11}a_{22}-a_{12}a_{21}\neq 0$. Is it true that |$\displaystyle a_{11}a_{22}-a_{12}a_{21}|\leq 1$?

I could prove that the above is true in assumption $\displaystyle 1>a_{ij}>0$. For other cases i tried to find counterexample with help of Maple but since the conditions are many Maple couldn't solve a system of equations with max and absolute value. Can anybody help me with this problem? Thank's in advance.
golomorf is offline  

  My Math Forum > College Math Forum > Linear Algebra

independent, inequality, max, vectors

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Triangle Inequality: Prove Absolute Value Inequality StillAlive Calculus 5 September 2nd, 2016 11:45 PM
If A, B independent events. Then A and B^C, as well as, A^C and B are independent beesee Probability and Statistics 2 June 9th, 2015 01:52 PM
independent of A rakmo Algebra 5 March 28th, 2013 06:48 AM
Function to generate linearly independent vectors. sparse_matrix Linear Algebra 0 November 8th, 2012 01:40 AM
Are X and Y independent. varunnayudu Advanced Statistics 2 November 27th, 2010 09:08 PM

Copyright © 2017 My Math Forum. All rights reserved.