Yahoo Answers is shutting down on May 4th, 2021 (Eastern Time) and beginning April 20th, 2021 (Eastern Time) the Yahoo Answers website will be 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
Linear Algebra Problem: Projections?
I don't really understand what this is saying, and how to prove it.
A linear mapping T: V -> V is called a projection if every element in w in the image is fixed by T. (That is, T(w) = w).
a) Let T be a projection. Prove that every v in V can be written as v = w+n with w in W and n in the nullspace of T.
b) Prove also that this expression is unique. That is, if v = w + n and v = w' + n' with w, w' in the image and n, n' in the nullspace, then w = w' and n = n'.
My professor pretty much through up the idea of the whole T: V -> V thing and I don't understand it at all especially where the w's come into play, and a little explanation of what exactly is going on would be appreciated.
Thanks
So the n and n' are = 0 because they are in the null space correct?
2 Answers
- 1 decade agoFavorite Answer
a) T(w+n)=(=-because of liner transform )=T(w)=>w+T(n)=>0==w+0=w.
b)T(w+n)=T(w'+n')
T(w)+T(n)=T(w')+T(n')
w+0=w'+0
w=w'
generally all transforms have an image and/or kernel ,and the her job is to moves vectors from subspaces or spaces w' is just a vector even if you don't know how it look like or you can call it x in f(x) so ( f==T )
