How could we prove that it is not a spanning set.

Question

Consider the space $\mathbb{R}$ as a linear space over the field $\mathbb{Q}$ of rational numbers. For any transcendental number x the set {1, $x$, $x^2$, $x^3$,......} is linearly independent. How could we prove that it is not a spanning set.

Answer 1

I imagine that you mean linearly independant over $\mathbb {Q}$. It is important to precise the field. Then you can invoke a cardinality argument, i.e. the cardinal of the span subspace of your set over $\mathbb {Q}$ is countable, while the set of real numbers is not.