Proof affine function as matrix equation

In summary: Since these are both in the same plane, the function is affine. Next, suppose that ##xeq 0##. Then, by your definition, the function is not affine anymore, since it would now be represented by f(x) = Ax + b, which is not an affine function.
  • #1
divB
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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?
 
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  • #2
divB said:

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.
 
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  • #3
divB said:

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##.
 
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1. What is an affine function?

An affine function is a type of mathematical function that maps a set of input values to a set of output values. It can be written in the form f(x) = mx + b, where m is the slope or rate of change and b is the y-intercept. Affine functions are commonly used in geometry and physics to describe linear transformations.

2. How can an affine function be represented as a matrix equation?

An affine function can be represented as a matrix equation by using a vector to represent the input values and a matrix to represent the linear transformation. The matrix will have the slope values as its coefficients and the vector will contain the input values. The resulting vector will then be added to the y-intercept to get the output values.

3. What is the purpose of proving an affine function as a matrix equation?

Proving an affine function as a matrix equation allows us to use linear algebra techniques to analyze the function. This can help us understand the properties and behavior of the function, as well as make it easier to solve for unknown values.

4. Can any affine function be represented as a matrix equation?

Yes, any affine function can be represented as a matrix equation. This is because all affine functions have a linear transformation component, which can be represented by a matrix, and a constant component, which can be represented by a vector.

5. Are there any limitations to representing an affine function as a matrix equation?

One limitation is that the matrix equation representation only works for affine functions with a single variable. It cannot be used for functions with multiple variables or for non-linear functions. Additionally, the matrix representation may not be practical for very complex or high-dimensional affine functions.

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