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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

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

    Ray Vickson

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    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##.
     
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