Prove the lemma by rephrasing If T cu dv cTu dTv for each c

Prove the lemma by rephrasing: If T( cu+ dv) = cT(u)+ dT(v) for each c,d in R and u v in R^n then T is a linear transformation.

idea: because the statement is indicated to be true (T is a linear transformation) then it must be true. ???

You must use your own words to rewrite the explanation above into s complete proof that makes sense to you.

Solution

Proof.

Set, d=0

So for any u in R^n and any c in R

T(cu)=cT(u)

Set, c=d=1

So for any u,v in R^n we have

T(u+v)=T(u)+T(v)

HEnce, T is a linear transformation