Self note on Linear maps of the following type: Let x in R and let a be constant, and let L be a linear map, then if L(a*x/|x|) < 1, then we have a*L(x/|x|) < 1, and multiplying by |x| on both sides, we have a*|x|*L(x/|x|) < |x|. Given linearity of L, however, we can write a*L(x/|x| +...{|x|times}...+x/|x|)= a*L(|x|*x/|x|) < |x|. Then a*L(x) < |x|. Any issues problems with this (normally one might expect |x| to be an integer in applying linearity here, or in other words when dealing with irrationals should there be any issues applying linearity in this manner?
Sunday, June 22, 2014
Self note on Linear Maps and a bit of algebra
Self note on Linear maps of the following type: Let x in R and let a be constant, and let L be a linear map, then if L(a*x/|x|) < 1, then we have a*L(x/|x|) < 1, and multiplying by |x| on both sides, we have a*|x|*L(x/|x|) < |x|. Given linearity of L, however, we can write a*L(x/|x| +...{|x|times}...+x/|x|)= a*L(|x|*x/|x|) < |x|. Then a*L(x) < |x|. Any issues problems with this (normally one might expect |x| to be an integer in applying linearity here, or in other words when dealing with irrationals should there be any issues applying linearity in this manner?
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