Prove affine mappings have form f(x)=Tx+b and properties

[TheRookie]

ℝ

Homework Statement

I'm a 2nd year undergraduate student, so I suppose many users here won't find this too difficult, but I've had some issues with the following questions and, of course, any help would be very much appreciated:

(i) Prove f: V→W is affine (where V and W are real vector space) iff it is of the form f(x) = Tx + b, where T: V→ W is linear and b∈W. Prove T and b are uniquely determined by f.

(ii) Prove the image of an affine subset under an affine mapping is affine. Prove the composition of two affine maps is affine.

Homework Equations

The Attempt at a Solution

(i) If f: V→W is an affine mapping, we must have f(λx + (1-λ)y) = λf(x) + (1-λ)f(y) for all x and y in V, and λ in ℝ.
This is what immediately came to mind, but I can't see a connection between this and the question...

(ii) I'm sorry, but I've gone over this again and again and don't know where to start.Regards,
Pete

 

[Bacle2]

Try substituting the condition on 3i) into f(x) in 1i), and see if it is satisfied. I think it may help you over the

long run if you have a good feel for the meaning of an Affine map. Not that you should

drop rigor at all, but it helps if you understand what is going on at an informal level.

Basically, to start, an affine map is a translation/shift of a linear map.

 

[micromass]

This thread has been closed because of academic misconduct.