Why is the trivial vector space the smallest vector space?

I suspect part of your confusion is,

what does "smallest" mean?

It seems to imply a partial ordering somehow, so here are two possible definitions:

  • $V$ is smaller than $W$ provided there is an injective linear map $V\to W$.
  • $V$ is smaller than $W$ provided $|V| \leq |W|$.

(Bonus questions: Is there any relationship between these definitions? Can one be proven from the other, and vice versa?)

Now, given either definition, say $V$ is the smallest vector space provided $V$ is smaller than $W$ for any vector space $W$.

From this definition, can you prove that $\{0\}$ is the smallest vector space? (Hint: Every vector space must have a $0$ element, so ...)

Tags:

Linear Algebra

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