For example.
- We know that a linear transformation is exactly the unique homomorphism between vector spaces, therefore applying a linear transformation to a vector always leads to a vector.
- At the same time, we know that the dual vector is the linear transformation that maps a vector to an element of the underlying field.
Combining the two, this means that the field element mapped from the dual vector is still a member of a vector space (which has to be one-dimensional, because it is a single element).
Can the field (underlying a vector space) always be considered as a uni-dimensional vector space?
Best Answer
A field $\mathbb{K}$ can always be considered to be a $\mathbb{K}$-vector space of dimension 1. It does not depend on the fact that there may pre-exist, or not, some other vector space for which $\mathbb{K}$ is the underlying field.
To prove this, it is enough to show one construction and there is one which is trivial: let $V = \mathbb{K}$ and let $ \cdot $ denote the internal product from $ \mathbb{K} \times \mathbb{K}$ to $\mathbb{K}$. Now define $*$, the external product from $ \mathbb{K} \times V $ to $V$, as being: $$k * v := k \cdot v$$ (let the external product be the internal one, disguised).
Then it is easy to show that $(V,+,*)$ is a 1-dimensional $\mathbb{K}$-vector space. No duals involved.