TITLE: Abusing the “Proportional To” symbol
QUESTION [5 upvotes]: As far as I know, the statement “$a \propto b$” is equivalent to “$a = kb$ for an arbitrary value $k$”. Is there anything wrong with the following?
$$
\begin{align}
f(x) &\propto 2\,f(x) \\
6 &\propto \pi \\
\begin{bmatrix} -5 \\ 10 \end{bmatrix} &\propto \begin{bmatrix} 1 \\ -2 \end{bmatrix} \\
A\mathbf{x} &\propto \mathbf{x}
\end{align}
$$
I want to use this in an informal setting and am wondering if it complete nonsense. Has this notation been (ab)used like this anywhere? Should I first clarify that “$a \propto b \Longleftrightarrow a = kb$ for an arbitrary $k$” at the top of the page?

(Other notations like “$a$ is a multiple of $b$ $\Longleftrightarrow$ $a|b$”, are too restrictive for my intended use because they sort-of imply integer multiplicity and can’t be chained together nicely, like $2(a, -a) = (2a, -2a) \propto (1, -1)$ for example.)
Thanks for your input.

Edit: I was asked to give some context: I’m just a student hand-writing his linear algebra homework. I come across myself wanting to emphasise that a vector is a scalar multiple of another in a string of equalities without having to write “where $a, a’ ∈ ℝ$” everywhere or writing words. Snippet:
$$
\mathbf{q}_2 = \mathbf{v}_2 - \text{Proj}_\mathbf{q1}\mathbf{v}_2
= \begin{bmatrix} 1 \\ -1 \end{bmatrix} + \frac{1}{5}\begin{bmatrix} 1 \\ 2 \end{bmatrix}
= \begin{bmatrix} \frac{6}{5} \\ -\frac{3}{5} \end{bmatrix}
\propto \begin{bmatrix} \frac{2}{\sqrt{5}} \\ -\frac{1}{\sqrt{5}} \end{bmatrix} = \mathbf{u_2}
$$
I realise it’s laziness, but I got curious. I thought it was pretty.

REPLY [2 votes]: To give a different viewpoint, there is actually a standard set which is the set of vectors up to scaling, and it is called $\mathbb{R}\mathrm{P}^n$ or $\mathbb{C}\mathrm{P}^n$, depending on the underlying scalar field.  They are pronounced "real projective space" and "complex projective space," respectively.
Since $\mathbb{R}\mathrm{P}^n$ is defined as a set of equivalence classes, where two nonzero vectors are in the same equivalence class if and only if they are scale multiples of each other, it would be fine to use the default equivalence symbol $\sim$, so long as you define what you mean.  For example,
\begin{equation}
\begin{pmatrix}1\\2\end{pmatrix}
\sim
\begin{pmatrix}2\\4\end{pmatrix}.
\end{equation}
(The thing I don't particularly like about $\propto$ is that it seems like a non-symmetric relation, though it is a clever use of the symbol, and if you define what you mean it is perfectly OK.  In fact, you might say $v\propto w$ iff $v=cw$ for some $c\in\mathbb{R}$, and then you are allowed to say $\mathbf{0}\propto \mathbf{x}$.   This is in contrast to $\mathbb{R}\mathrm{P}^n$ where we do not allow comparison with the zero vector.)