Proof affine function as matrix equation

divB

Homework Statement

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

Homework Equations

Affine function: $f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)$ with $\alpha+\beta=1$

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?

Homework Helper

Homework Statement

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

Homework Equations

Affine function: $f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)$ with $\alpha+\beta=1$

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.

1 person
Homework Helper
Dearly Missed

Homework Statement

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

Homework Equations

Affine function: $f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)$ with $\alpha+\beta=1$

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##.

1 person