Proof affine function as matrix equation

  • Thread starter divB
  • Start date
  • #1
87
0

Homework Statement



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]

Homework Equations



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

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?
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,260
619

Homework Statement



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]

Homework Equations



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

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.
 
  • #3
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722

Homework Statement



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]

Homework Equations



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

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##.
 

Related Threads on Proof affine function as matrix equation

Replies
5
Views
599
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
2
Views
11K
Top