1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Proof affine function as matrix equation

  1. Sep 28, 2013 #1
    1. The problem statement, all variables and given/known data

    Proof that any affine function can be written as [itex]f(x) = Ax + b[/itex], [itex]A \in \mathbb{R}^{m\times n}[/itex] and [itex]x,y \in \mathbb{R}^n[/itex], [itex]b \in \mathbb{R}^m[/itex]

    2. Relevant equations

    Affine function: [itex]f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)[/itex] with [itex]\alpha+\beta=1[/itex]

    3. The attempt at a solution

    I could proof that the function f(x)=Ax + b is affine.

    However, I am stuck proofing that any affine function can be represented so.
    Any pointer how I can start here?
     
  2. jcsd
  3. Sep 28, 2013 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Define the function g(x)=f(x)-f(0) and try to prove g is linear.
     
  4. Sep 28, 2013 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    The word you want is 'prove', not proof. To prove something is to supply a proof.

    Anyway, to start, apply your definition of "affine" to the case of ##x \in \mathbb{R}^n## and ## y = 0 \in \mathbb{R}^n##.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted