Quote from Lecture 3, 6.S897 Algebra and Computation by Madhu Sudan, MIT:
Suppose we have a vector space $(V,+,\circ)$ where:
- $V$ is a set of vectors
- $+$ is vector addition operator: $V \times V \mapsto V$
- E.g. $\vec a + \vec b = \vec c$
- $\circ$ is vector scaling operator: $F \times V \mapsto V$
- E.g. $c \circ \vec v = \vec u$
Note that, dot product $\cdot$ and cross product $\times$ are NOT part of the vector space definition!
We don’t have to follow the notation in the note above, so, taking the definition of Rings, we have
- Case 1: $(V, +)$ corresponds to $(R, +)$ and $(V, \circ)$ to $(R, \cdot)$
- Case 2: $(V, +)$ corresponds to $(R, \cdot)$ and $(V, \circ)$ to $(R, +)$
- $(V, +)$ is an Abelian group. (satisfies condition 1)
- $(V, \circ)$ is NOT a monoid. (fails condition 2)
- $(V, \circ)$ is NOT a monoid. (fails condition 1)
So a vector space is not even a ring. Of course it cannot be a field.
A side dish of this post is that we now know $(V, +)$ is an Abelian group.
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