Is the Affine Transformation T(x) = Ax + y a Linear Transformation?

[tnutty]

Homework Statement

An affine transformation T : R^n --> R^m has the form T(x) = Ax + y, where x,y are vectors
and A is an m x n matrix and y is in R^m. Show that T is not a linear transformation when
b != 0.

Homework Equations

There are couple of useful definitions, but I think this one will suffice :

Definition :

If T is a linear transformation, then T(0) = 0

and T(cU + dV) = cT(U) + dT(V), for all vectors U,V in the domain of T and all scalars c,d.

The Attempt at a Solution

I figure I would show that the affine transformation does not satisfy the definition.

Attempt :

Let U,V be vectors, and c,d be scalers.

Then the affine transformation has to satisfy T(cU + dV) ?= cT(U) + dT(V).

since T(x) = Ax + y then

T(cU + dV) = A(cU + dV) + y

= cAU + dAV + y

= cAU + dT(V)

thus cAU + dT(V) != cT(U) + dT(V), which means that affine transformation is not a
linear transformation.

Would this suffice or Do I make some assumption that are not plausible. BTW, there
was a theorem that said that If A is an m x n matrix, then the transformation x -> Ax has
the properties A(U + V) = AU + aV and A(sU) = sA(U), where U,V are vectors in R^n
and s is a scaler.

 

[owlpride]

Yes, it suffices to show that T(cU + dV) =!= cT(U) + dT(V). However, there's an easier way:

If T(x) = Ax + y, then T(0) = y =!= 0 unless y=0.

 

[tnutty]

Oh, I should have read the question carefully. Thanks a lot.