As I was self-studying the geometric intuition behind the exterior and tensor algebras, I naturally decided to review covariant and contravariant vectors. Of course, this led me back to vector spaces and dual spaces whereupon I stumbled upon a question.
Let $V$ be a finite-dimensional vector space over a field F. Let $V*$ be its dual space.
Could we define vector/covector addition and scaling using relations? Note that I am using the definition of a relation as a subset of a cartesian product between elements of either V, V*, or F.
Here is what I am thinking. Note that I will not be using the conventions regarding upper/lower indices for the purposes of this question.
Define vector addition as a relation as follows. Fix a $w \in V$. Then, $\forall u, w \in V$, we say that $ v+u = w$ iff $v_i + u_i = w_i$, where $v_i, u_i, w_i$ are the $i$th scalar components of $v, u, w$ respectively with respect to the basis of the vector space.
Similarly, define vector scaling as a relation as follows. Fix a $u \in V$. Then, $\forall v \in V$ and $ \forall \lambda \in F$, we say that $\lambda v = u$ iff $\lambda v_i = u_i$.
Define covector addition as a relation as follows: Fix a $ \theta \in V*$. Then, $\forall \psi, \phi \in V*$ and $\forall x \in V$, we say that $ \psi+\phi = \theta$ iff $\psi(x)+\phi(x) = \theta(x)$.
Similarly, define covector scaling as a relation as follows: Fix a $\theta \in V*$. Then, $ \forall \psi \in V*$, $ \forall x \in V$, and $\forall \lambda \in F$, we say that $ \lambda \psi = \theta$ iff $\lambda \psi(x)= \theta(x)$.
Do these definitions make sense? Are they consistent with the usual definition of vector/covector operations as maps from cartesian products to a single element of $V$ or $V*$?
If these definitions do work, then I think that the usual axioms regarding vector spaces and dual spaces can be derived. It is also possible that I am completely wrong here. Either way, any guidance would be appreciated. Thanks in advance!
The definition on the dual space is correct, though we don't need to express the addition and scalar multiplication functions as relations.
For the vectors, you're wrong: there's no such thing a priori as the '$i$th scalar component' of a general vector in a general vector space, only after (finding and) fixing a basis.
Your work, however, is basically just showing that $F^n$ with the coordinatewise operations satisfies the definition of a vector space.