# Prove this vector space is not finite-dimensional.

• Feb 13th 2010, 08:41 PM
tianye
Prove this vector space is not finite-dimensional.
Let V be the set of real numbers.Regard V as a vector space over the field of rational numbers,with the usual operations.Prove that this vector space is not finite-dimensional.
It is easy to give countable real numbers which are linearly independent intuitively.But I always fail to prove this.
Can you help me?Thanks.
• Feb 14th 2010, 04:38 AM
tonio
Quote:

Originally Posted by tianye
Let V be the set of real numbers.Regard V as a vector space over the field of rational numbers,with the usual operations.Prove that this vector space is not finite-dimensional.
It is easy to give countable real numbers which are linearly independent intuitively.But I always fail to prove this.
Can you help me?Thanks.

Using the fact that $\displaystyle \pi$ is a transcendental number , show that the set $\displaystyle \{1,\pi,\pi^2,\ldots,\pi^n\}$ is linearly independent over $\displaystyle \mathbb{Q}\,\,\,\forall\,n\in\mathbb{N}$

Tonio