# Prove that a function is affine if and only if it is of the form f(x)=Tx+b

1. Mar 11, 2012

### gilabert1985

Hi, I have the following problem that is part of a project, and I have been stuck on it for the last couple of hours ... Thanks a lot for any help you can give me!

It says:

"An affine mapping from V to W, where W is a second real vector space, is a mapping f:V$\rightarrow$W such that

for all x,y$\epsilon$V and all $\lambda\epsilon\Re$.

QUESTION:
Prove that f:V$\rightarrow$W is affine if and only if it is of the form

f(x)=Tx+b, ---> (this is formula (1))

where T:V$\rightarrow$W is linear and b$\epsilon$W. [Hint: to prove the 'only if' part, consider the mapping f-f(0).]

Show further that T and b in (1) are uniquely determined by f."

So I know the definition of affine functions from Wikipedia (http://en.wikipedia.org/wiki/Affine_transformation), so the hint I get that suggests me to consider the mapping f-f(0) makes sense. However, I have no clue how to do it without numbers (like it is done in the examples of page 7 and 8 of these notes http://cfsv.synechism.org/c1/sec15.pdf)

2. Mar 12, 2012

### micromass

Staff Emeritus