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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?
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    I don't understand what it means to take this union.2011-07-07
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    @Qiaochu Yuan: $E$ is the set of free vectors which is the union of bound vectors.2011-07-07
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    @Damien: the union of all these vectors can actually be made into a manifold; the collection of vectors based at a fixed point O is actually a vector space ( of dimension n; the same as that of $R^n$)usually called the tangent space of $R^n$, based at O; the union of all these spaces also has a name, and . Look up Tangent Bundle. BTW, tangent spaces based at any two different points are naturally isomorphic to each other; basically by translating vectors to the origin and then back.2011-07-07

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