Preferred notation for a vector starting with $1$'s

Question

Is there a commonly accepted notation for a vector whose first $k$ entries are $1$'s, with $0$'s afterwards? I have seen $\mathbf{e}_i$ for a vector with a $1$ in the $i$th entry, and $\mathbf{e}$ for a vector of all ones, but not this particular case.

Answer 1

I don't think so, but if you want to write things and use these vector all the time, just define $v_k=\sum_{i=1}^k e_i$. Or you might define $e_S=\sum_{i\in S}e_i$ (which seems a reasonable extension for the notation), so the vectors you are looking for would be $e_{I_k}$.

Answer 2

To my knowledge there is no standard notation for this.

Since it is standard to use $\mathbf{e}_i$ to refer to the $i$th column of the identity matrix $I_n$, I would write something like

Let $\mathbf{1}_k$ be the vector whose first $k$ entries are $1$ and whose last $n-k$ entries are $0$. Note that $\mathbf{1}_k=\mathbf{e}_1+\mathbf{e}_2+\dotsb+\mathbf{e}_k$.

If you prefer more descriptive notation, then I think Chris Cutler's suggestion to write $\mathbf{e}_{[1,k]}$ to refer to this vector is a good idea. I think $\mathbf{1}_{\leq k}$ would work well too.