What does it mean for to be a vector space over $\mathbb R$ or $\mathbb C$? Are they fields?
I have a list of subsets (like matrices and linear maps) and am trying to determine if they are vector spaces over R and/or over $\mathbb C$, but I'm not sure how to do this.
I know vector spaces need to be subspaces so I have been trying to use the Subspace Test, but I have no idea if that works for both "over $\mathbb R$ and/or over $\mathbb C$"?
Thanks!
As part of the definition of a vector space, there is a scaling action, that takes a scalar and a vector, and produces a new vector. These scalars most come from somewhere.
This somewhere is a field, and the vector space is said to be a vector space over that field. Thus "$V$ is a vector space over $K$" means that $V$ is a vector space that allows scalars from the field $K$.
So the set of $2\times 2$ matrices with real entries and standard addition is easily seen to be a vector space over the real numbers with standard scaling. But it is not a vector space over the complex numbers with standard scaling, as for instance, the identity matrix scaled by $i$ doesn't have real entries.
In contrast, the set of $2\times 2$ matrices with complex entries (and the standard operations) is both a vector space (of dimension 8) over the real numbers and a vector space (of dimension 4) over the complex numbers. In fact, any vector space over $\Bbb C$ automatically becomes a vector space over $\Bbb R$ with double the dimension in a very natural way.