Originally Posted by

**ymar** Let $\displaystyle u_1,...,u_n\in\{0,1\}^n$. Can we impose a condition on these vectors so that every vector $\displaystyle s\in\{0,1\}^n$ has a unique decomposition

$\displaystyle s=\alpha_1\cdot u_1\vee ... \vee\alpha_n\cdot u_n,$

where $\displaystyle \alpha_i\in\{0,1\},$ and by definition $\displaystyle \alpha\cdot \begin{pmatrix}x_1\\\vdots\\x_n\end{pmatrix}=$$\displaystyle \begin{pmatrix}\alpha\wedge x_1\\\vdots\\\alpha\wedge x_n\end{pmatrix}$ and $\displaystyle \vee$ denotes element-wise conjunction?

The one from linear algebra (a linear combination can be zero only trivially) doesn't work -- it's easy to give an example.