Proof affine function as matrix equation

divB · Sep 28, 2013

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?

Dick · Sep 28, 2013

divB said: ↑

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?

Define the function g(x)=f(x)-f(0) and try to prove g is linear.

Ray Vickson · Sep 28, 2013

divB said: ↑

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?

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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Proof affine function as matrix equation

divB

Dick

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