Extension of the field of a matrix vector space

Solution 1:

The answer is yes. In the most general setting, extending the field of scalars is done via the (appropriately named) extension of scalars. Of particular note is the notion of complexification. Note also that a "complexified" real vector space is sometimes referred to as a linear complex structure.

Whatever your field $F$, it is not too difficult to establish that there exists a sensible notion of multiplication by the extended field $F'$. The "tricky" parts are usually related to how the original vector space "fits" within this extended structure. For example: the complex eigenvalues and eigenvectors of a real matrix come in complex-conjugate pairs; coming up with (and proving) an analogous statement for endomorphisms over a complexified vector space means that we need to say what exactly "having real entries" should mean in the abstracted context.