Do either of the linear transformation properties imply the other? Im

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Do either of the linear transformation properties imply the other?  Im

Amiya Wolf · Accepted Answer

Consider two vector spaces V, V&#039; and a map T:V→V′. If for any v1,v2∈V,T(v1+v2)=Tv1+Tv2, then ∀n∈Z,T(nv1)=Tv1+⋯+Tv1=nTv1, and hence ∀α∈Q,v∈V,T(αv)=αTv.Given any irrational number 0&amp;lt;r&amp;lt;1, we can write r=∑k=1∞10−krk. That is, ∀ϵ&amp;gt;0,∃N∈N such that every positive integer n≥N gives 0&amp;lt;r−∑k=1n10−krk&amp;lt;ϵ. In this case, ∀v∈V,rTv&amp;gt;T(∑k=1n10−krkv)=Tv∑k=1n10−krk⋅&amp;gt;(r−ϵ)Tv.This implies T(rv)=rT(v)∀r∈R.For the other direction, ∀v1,v2∈V,T(v1+v2)=T(v1(1+v2v1)), provided v1,v2=(1+v2v1)Tv1=Tv1+Tv2are scalar multiples of each other or V is 1 dimensional.Note that given T(1,0,0) and T(0,0,1), it is impossible to find T(1,0,1) as (1,0,1) is not a scalar multiple of either (1,0,0) or (0,0,1).

Do either of the linear transformation properties imply the other? I'm new to linear algebra and

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