(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant 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.

3. 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Affine transformation proof

**Physics Forums | Science Articles, Homework Help, Discussion**