One example I like to use is the $1$-dimensional vector space of multiples of some physical unit (length, time, mass): for example, the *meter* is a basis of the $1$-dimensional vector space of *lengths*, and the *light-year* is also a basis of it, but there is no *natural* basis of this vector space.

This example can also be used to illustrate multilinear algebra constructs on $1$-dimensional vector spaces: the space of *speeds* is the (still $1$-dimensional) space of linear maps between the $1$-dimensional vector space of time spans and the $1$-dimensional vector space of lengths (it turns out that, in special relativity, but not in classical mechanics, there is a canonical isomorphism between these spaces, i.e., a canonical basis for the space of speeds).  The space of *areas* it the tensor square of the space of lengths, and the space of *volumes* is its tensor cube.  And so on.

This kind of example makes it clear why for $1$-dimensional vector spaces $V$, the tensor product of $V$ with its dual is *canonically* isomorphic to the base field, so such spaces can be called "invertible" (as in "invertible sheaf").