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.
Best Answer
This usage of proportionality is unusual. Two "proportional" vectors are also parallel, hence the parallel symbol is more suitable for vectors $$ \begin{bmatrix} -5 \\ 10 \end{bmatrix} \parallel \begin{bmatrix} 1 \\ -2 \end{bmatrix} $$ $$ A\mathbf{x} \parallel \mathbf{x} $$ Functions are essentially vectors so $f \parallel g$ is also suitable.