- #1

psholtz

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## Homework Statement

I'm trying to show that every affine function f can be expressed as:

[tex]f(x) = Ax + b[/tex]

where b is a constant vector, and A a linear transformation.

Here an "affine" function is one defined as possessing the property:

[tex]f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)[/tex]

provided that:

[tex]\alpha + \beta = 1[/tex]

## The Attempt at a Solution

I've defined:

[tex]g(x) = f(x) - f(0)[/tex]

and the idea is to show that g(x) is linear. If so, the form of f we are trying to derive above follows easily.

It's easy to show that g maps zero onto zero:

[tex]g(0) = f(0) - f(0) = 0[/tex]

and it's easy to show that:

[tex]g(\alpha x) = f(\alpha x) - f(0)[/tex]

[tex]g(\alpha x) = f(\alpha x + (1-\alpha) \cdot 0) - f(0)[/tex]

[tex]g(\alpha x) = \alpha \cdot f(x) + (1-\alpha)\cdot f(0) - f(0)[/tex]

[tex]g(\alpha x) = \alpha \cdot f(x) - \alpha \cdot f(0)[/tex]

[tex]g(\alpha x) = \alpha \cdot \left( f(x) - f(0) \right)[/tex]

[tex]g(\alpha x) = \alpha \cdot g(x)[/tex]

But I'm having more trouble proving the property:

[tex]g(x+y) = g(x) + g(y)[/tex]

On the one hand we have:

[tex]g(x+y) = f(x+y) - f(0)[/tex]

and on the other hand we have:

[tex]g(x) + g(y) = f(x) + f(y) - 2f(0)[/tex]

so it seems that if we could prove that:

[tex]f(x + y) = f(x) + f(y) - f(0)[/tex]

we would be done.

This relation seems to hold for various affine functions that I've tried substituting into it, but I'm having trouble proving it in general.