I have a general question about vectors and Euclidean space:
Suppose we are working in $\mathbb{R}^3$. In this space we can identify points. Choose an arbitrary point and label it $O$. Then we can consider the vector space $V_{O}$ of displacement vectors. Then consider $E = \bigcup_{A \in \mathbb{R}^3} V_A$
Note that $E$ is the set of free vectors which we can write as the union of bound vectors over points $A \in \mathbb{R}^3$.
Is $E$ just a union of vector spaces? More specifically:
- In looking at $\mathbb{R}^3$ we transition from points to displacement vectors with some fixed origin $O$. Consider a vector $\textbf{v}$ and its parallel copies with respect to $O$. Then this is an equivalence class. Is this where the identification ends? Or should we also consider other equivalence classes with different origins? The set of all of the equivalence classes is forms a quotient space?