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 + ry) = T[x) + rT(y) for all x, y in R^m and r epsilon R.

PRoblem 5

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

Let, x=y=0 and r=1

So we get

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

Hence, T(0)=0

Set, r=1

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

Set, x=0 we get

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

Hence , T is linear

Assume T is linear

Hence,

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

Hence proved