I'm coming back to this [question](https://mathoverflow.net/questions/182621/the-space-of-sequences-of-rationals-and-its-dimension). Is it possible to have "an explicit" linearly independent family of sequences of rationals with a cardinal equal to the continuum?


PS: sorry for the duplicate on the previous question.



Linearly independent family of sequences of rationals with a cardinal equal to the continuum
Choose a bijection $\alpha:\mathbb{N}\to\mathbb{Q}$, and for each $x\in\mathbb{R}$ let 
$$a(x)\_i=\begin{cases}0&\mbox{ if $\alpha(i)<x$}\\1&\mbox{ if $\alpha(i)\geq x$}\end{cases}.$$
Then the set of sequences $\{a(x):x\in\mathbb{R}\}$ is linearly independent.


