Prove that a transformation T from Rm to Rn is linear if and

Prove that a transformation T from R^m to R^n is linear if and only if T(x vector + ry vector) = T(x vector) + rT(y vector) for all x vector, y vector in R^m and r elementof R.

Solution

Let T be linear

Then,

T(x+ry)=T(x)+T(ry)=T(x)+rT(y)

Let,

T(x+y)=T(x)+rT(y)

Set, r=1

So, T(x+y)=T(x)+T(y)

Now let ,r=1 and x=y=0

T(0+1*0)=T(0)+1*T(0)=2T(0)

Hence, T(0)=0

So we now set x=0

So, T(0+ry)=T(0)+rT(y)

T(ry)=0+rT(y)=rT(y)

Hence, T is linear

Hence proved