Re: How to define a vector without coordinate system?

R^1 is one-dimensional vector space.

Definition
Let $F$ be a field. A {\em vector space} $V$ over $F$ is a set with two
operations, $+: V \times V \longrightarrow V$ and $\cdot: F \times V
\longrightarrow V$, such that:
etc.(see details here)
http://planetmath.org/encyclopedia/VectorSpace.html

If you choose field F=R and set V=R then you will get one-dimensional
vector space. We can choose basis vector for example 1 (it can be any
other number).

Sqrt(2) = Sqrt(2)*1
Sqrt(3) = Sqrt(3)*1

Sqrt(3) = a*Sqrt(2)

where a=(Sqrt(3)/Sqrt(2));
Then Sqrt(3) and Sqrt(2) are linearly dependent.

The same operation with with Pi.

Pi=a*1

where a=Pi.

Andrzej

.

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