# Is there something wrong with this proof that a composition of affine linear function is affine linear

I'm trying to write what I think is a very simple proof for my master's thesis. I know this has already been proved, but I wanted to try to prove it myself to see what are the shortcomings of my understanding.

I want to prove that a composition of affine linear functions of form $$f(x) = a \cdot x + b$$ is also an affine linear function of the same form.

What I have so far looks like this :

$$f_1(x) = a \cdot x + b$$

$$f_2(x) = c \cdot x + d$$

$$f_2(f_1(x)) = c \cdot (a \cdot x + b) + d$$

$$= c \cdot a \cdot x + c \cdot b + d$$

$$= (c \cdot a) \cdot x + (c \cdot b+d)$$

Substite $$(c \cdot a)$$ and $$(c \cdot b + d)$$ by some constants and you get an equation of form $$f(x) = a \cdot x + b$$.

Is there something wrong with this approach? I think it's pretty straight forward, but I want to see if there's something I misunderstood.

• Use \cdot for multiplication, or no symbol at all. * or \ast usually denotes different things, and looks strange for multiplication Commented Jan 17, 2022 at 21:31
• Usually, especially when one is speaking of "linear-algera", as you have tagged this. ,linear means that $F(a+b)=F(a)+F(b)$ and $F(\lambda a)=\lambda F(a)$ for scalars $\lambda$.
– lulu
Commented Jan 17, 2022 at 21:32
• Whether that's acceptable depends on the context. As an official writeup of an official proof it's meaningless. You need some English in there to clarify which equations are things we're assuming and which are consequences of the assumptions. Students often don't believe they can't write a proof with no English, which is curious, since every proof in their texts is a sequence of English sentences. Commented Jan 17, 2022 at 21:32
• You have a typo in the definition of $f_2$. How would there be any other approach to this (other than, say, using matrices)? Commented Jan 17, 2022 at 21:38
• Should say: this all seems a bt odd and unclear. If you are speaking of linear algebra, as you have tagged, then nothing in your "proof" is at all relevant. If you tagged linear algebra by accident, somehow, and you really just meant functions of this form, then you could clean this up to make an argument. But, is that really what you meant? Seems like an odd thing to have in a master's thesis.
– lulu
Commented Jan 17, 2022 at 22:12

### Style

The goal of a proof is to convey an idea to an audience. Typically, this is done by mixing both notation and English. If you try to read old mathematics (e.g. Babylonian or Greek mathematics), you will find that they wrote everything out in words—modern notation didn't exist yet. This is quite difficult to parse, as the plain language needs to be interpreted and understood as mathematics. On the opposite end of the spectrum are formal proofs (e.g. those produced by and for automated proving systems). These tend to be a compressed muddle of pure notation, which is also quite difficult to read.

The goal is to come to a happy medium, where your principle tool is plain language, but where notation is introduced when it aids clarity. Everything should be written in complete sentences, with appropriate punctuation, spelling, grammar, and so on.

I would also suggest that you avoid pronouns. While a lot of authors will write using the "mathematical 'we'", I find this to be a little precious, and you should absolutely avoid the second person. My preference is to write in a more imperative style. For example:

$$\begin{array}. \color{red}{✘} & \text{"After substituting, you get an equation of the form f(x) = ax+b."} \\ \color{red}{✘} & \text{"After substituting, we get an equation of the form f(x) = ax+b."} \\ \color{green}{\checkmark} & \text{"Substitute to get an equation of the form f(x) = ax+b."} \\ \end{array}$$

### Definitions & Notation

As several commenters have pointed out, the functions you define as "linear" are not, in a more general setting, "linear functions". In introductory level classes, these kinds of functions are often called linear because they are lines in the plane, but, generally speaking, a function $$f$$ is linear if $$f(x+y) = f(x) + f(y) \qquad\text{and}\qquad f(\lambda x) = \lambda f(x)$$ for all appropriate values of $$x$$, $$y$$, and $$\lambda$$ (essentially, linear functions "distribute" over sums, and scalars can be "factored out" of linear functions).

You have two options here: (1) you can either adopt a non-standard definition and very clearly indicate that you are doing so, or (2) you can use standard definitions.

You have also used a number of variables in your argument which are not defined. What are $$a$$, $$b$$, $$c$$, and $$d$$? What are $$f_1$$ and $$f_2$$? have you defined them via the formulae at the top of your argument, or is this some derived property? There is something missing in the argument there.

There is also some confusion for me here regarding domains. I am assuming that $$f_1$$ and $$f_2$$ are functions which take the reals to the reals. However, this is neither obvious, nor necessary. You need to be very careful, in your definitions or exposition, to explain what the domains and codomains of your functions are.

Finally, all of those \cdots are distracting. I would get rid of them.

### Alternative Presentation

Acknowledging that I am on the verbose end of the spectrum when it comes to mathematical writing, my own presentation of the proposition in the question would be something like the following:

Definition: A function $$f : \mathbb{R}\to\mathbb{R}$$ is affine linear if there are real constants $$a$$ and $$b$$ such that for any $$x\in\mathbb{R}$$, $$f(x) = ax + b.$$

Proposition: The composition of two affine linear functions is affine linear.

Proof: Let $$f_1 : \mathbb{R}\to \mathbb{R}$$ and $$f_2 : \mathbb{R}\to \mathbb{R}$$ be two arbitrary affine linear functions. That is, suppose that there are real constants $$a_1$$, $$a_2$$, $$b_1$$, and $$b_2$$ such that $$f_1(x) = a_1 x + b_1 \qquad\text{and}\qquad f_2(x) = a_2 x + b_2.$$ Compose $$f_1$$ with $$f_2$$ to get \begin{align} (f_1 \circ f_2)(x) &= f_1(f_2(x)) \\ &= a_1 (a_2 x + b_2) + b_1 \\ &= (a_1a_2)x + (a_1b_2 + b_1). \end{align} Both $$a_1a_2$$ and $$a_1b_2 + b_1$$ are real constants, hence $$f_1 \circ f_2$$ is an affine linear map. $$\square$$