Prove a transformation T is linear if and only if Tc1v1cnvn

Prove a transformation T is linear if and only if T(c₁v₁+...+c_nv_n) = c₁T(v₁)+..c_nT(v_n) for all v₁...v_n in the domain T and for all scalars c₁...c_n

Solution

Let,T be linear

We prove by Induction

Base case n=1

T(c1v1)=c1T(v1) because T is linear.So base case is true

Inductive step: Assume true for some n>=1

We show it is true for n+1

T(c1v1+cnvn+cn+1vn+1)=T((c1v1+....cnvn)+cn+1vn+1)=T(c1v1+...+cnvn)+T(cn+1vn+1)

                          =c1T(v1)+...+cnT(vn)+cn+1T(vn+1)

Using induction hypothesis and fact taht T is linear

HEnce true for all n>=1

Now assume:

T(c1v1+...+cnvn)=c1 T(v1)+...cnT(vn)

Let, c1=c and ci=0 for all i>!

So, we get

T(c1v1)=c1 T(v1)

THis is true for any scalar c1 and any v1 in domain

Set, c1=c2=1 and ci=0 for all i>2

T(v1+v2)=T(v1)+T(v2)

Hence, T is linear transformation