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YCor
YCor
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The question is as in the titleLet $a_1, \dots, a_n$ be a finite set of positive reals. Is there a $\mathbb Q$-basis of $\mathbb R$ where each $a_i$ has nonnegative coordinates?

Playing around with the case $n = 2$, I’m pretty sure the answer is yes, but it seems a bit fiddly, and I think this should be well-known, with probably some good way to think about it that I’m missing.

The question is as in the title. Playing around with the case $n = 2$, I’m pretty sure the answer is yes, but it seems a bit fiddly, and I think this should be well-known, with probably some good way to think about it that I’m missing.

Let $a_1, \dots, a_n$ be a finite set of positive reals. Is there a $\mathbb Q$-basis of $\mathbb R$ where each $a_i$ has nonnegative coordinates?

Playing around with the case $n = 2$, I’m pretty sure the answer is yes, but it seems a bit fiddly, and I think this should be well-known, with probably some good way to think about it that I’m missing.

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Tim Campion
Tim Campion
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Let $a_1, \dots, a_n$ be a finite set of positive reals. Is there a $\mathbb Q$-basis of $\mathbb R$ where each $a_i$ has nonnegative coordinates?

The question is as in the title. Playing around with the case $n = 2$, I’m pretty sure the answer is yes, but it seems a bit fiddly, and I think this should be well-known, with probably some good way to think about it that I’m missing.