Yahoo Answers is shutting down on May 4th, 2021 (Eastern Time) and the Yahoo Answers website is now in read-only mode. There will be no changes to other Yahoo properties or services, or your Yahoo account. You can find more information about the Yahoo Answers shutdown and how to download your data on this help page.
Trending News
Prove that vectors x~y if xⁿ=yⁿ for all n=1...m?
Let x and y be vectors in m-dimensional real space and x~y denotes that x and y have the same coordinates (not necessarily in the same order). Let xⁿ denote the sum of n'th powers of the coordinates xⁿ = (x1)ⁿ+...+(xm)ⁿ where x1,...,xm denotes the m coordinates of vector x. Prove that x~y if xⁿ=yⁿ for all n=1,...,m.
This is a follow up on this question:
http://answers.yahoo.com/question/index;_ylt=Aqb2e...
Note that if x1,...,xm has no repetitions then there will be m! distinct vectors that satisfy x~y. Intuitively this fits perfectly with the fact that xⁿ=yⁿ is an n'th degree polynomial of some kind and solving it for one coordinate, say x1, will yield at most n solutions. In the line of this thought the system of m equations xⁿ=yⁿ for n=1,...,m might be shown to have at most m! solutions...
If you can show that the system has at most m! solutions we are done proving it for vectors with no repetitions in the list x1,...,xm. My intuition tells me that then it will be simple to extend it to vectors with possible duplicate coordinates.
Just for the record this is a repost as I made a technical mistake and spammed my first version of the question which I will soon delete:
2 Answers
- gianlinoLv 79 years agoFavorite Answer
The symmetric functions of the x_i's can be expressed in terms of the x^m. or the y^m
http://en.wikipedia.org/wiki/Power_sum_symmetric_p...
Therefore they coïncide with those of the y_i's. Hence the x_i's and the y_i's are both set of solutions of a single polynomial.
So x ~ y.
- anordtugLv 69 years ago
Guess ~ means equal in numerical value and direction.
As X and Y consist of equal vectors in different orders.
As X+Y= Y+X and XY=YX also for vectors, your mentioned equality should be obvious.