What kind of vector spaces have exactly one basis?

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Here is the question as an exercise in the book Linear Algebra Done Right, Chapter 2

Find all vector spaces that have exactly one basis.

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1 Answer

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If $\{v_1,v_2,\dotsc,v_n\}$ is a basis for a vector space $V$, then $\{v_1+v_2,v_2,\dotsc,v_n\}$ is also a basis.

So $V$ should have a basis of one element $v$, now for some nonzero and non-unit element $c$ of the field choose the basis $cv$ for $V$.

So $V$ must be a vector space with dimension one on a field isomorphic to $\mathbb Z_2$.

All vector spaces of this kind are of the form $V=\{0,v\}$ or the trivial one.

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