Showing that a function is affine in $\mathbb{R}^{n}$

Question

I'm reading an analysis book, and I'm struggling with a question. First, here is the definition for an affine function:

A function is said to be affine if it is defined by

$$g(u) = c + \sum_{i=1}^{n}a_{i}u_{i}$$

where $c$ and the $a_{i}'s$ are prescribed values. If $c = 0,$ the function is linear.

Now, here's the problem:

Fix a point $x$ in $\mathbb{R}^{n}$ and let $c$ be a point in $\mathbb{R}^{n}$ and define $\phi : \mathbb{R}^{n} : \mathbb{R}$ by $$\phi(u) = \langle c, u - x \rangle .$$

(a) Show that $\phi : \mathbb{R}^{n} \rightarrow \mathbb{R}$ is affine.

(b) Show that given any nonconstant affine function $\phi : \mathbb{R}^{n} \rightarrow \mathbb{R}$, it is possible to choose points $x, c \in \mathbb{R}^{n}$ so that $\phi : \mathbb{R}^{n} \rightarrow \mathbb{R}$ has the above form.

For (a), I used the bilinearity property of the inner product as follows:

$$\langle c, u - x \rangle = \sum_{i=1}^{n} c_{i}u_{i} - \sum_{i=1}^{n} c_{i}x_{i}.$$

I don't know how to get it into the form used in the definition, though. I also don't know how to get an arbitrary affine function into the form provided above. I'm guessing you start with $c + \sum_{i=1}^{n} a_{i}u_{i}$ and manipulate it into what we want. I tried a lot of things, but I couldn't get it.

Any help is appreciated.

Answer 1

For b) take $x_0=(a_1,a_2,...,a_n)$ and $x =-\frac c {\|(a_1,a_2,...,a_n)\|^{2}} x_0$. Then $\langle x_0, (u-x) \rangle =c+\sum a_i u_i$.

Answer 2

The term $-\sum_{i=1}^{n} c_ix_i$ is just a constant since $<c,u-x>$ is only a function of $u$. So, it does have the form of the affine function in the definition.

Part (b) follows from the fact that any number in $\mathbb{R}$ can be written as a linear combination of other numbers in $\mathbb{R}$ ($\mathbb{R}$ is closed under addition and scalar multiplication).