Can the scalars of a vector space be of a different field than the entries of the vector space? When denoting a particular vector space can the field that denotes the entries in the vectors of a vector space be different than the field that denotes the scalars that can be used on that vector space. For example if $V$ is a vector space denoted by $M_{m\times n}(R)$ do the scalars $c$ that act on that vector space have to be elements of $R$? I was reviewing the text "Linear Algebra" by Friedberg, Insel and Spence, and was a little confused by the following question:

Let $V$ denote the set of all $m\times n$ matrices with real entries; so $V$ is a vector space over  $R$ by example 2 (note this was the general definition of matrices over $R$). Let $F$ be the field of rational numbers. Is $V$ a vector space over $F$ with the usual definitions of matrix addition and scalar multiplication?

If $c\in R$ then no because you could do scalar multiplication with an irrational number and that would take you out of the vector space over $Q$ because an irrational number times a rational number is an irrational number. Now if $c$ is in $Q$ then yes it is a vector space. Without $c$ being explicitly denoted do I assume it belongs to the field that was defined with the vector space? Also, can $c$ and $V$ be different fields when denoting a vector space?
Yes, it is a vector space over F. 

A vector space over F is a set V (in your case $m\times n$ matrices with real entries) with operations $+:V\times V \to V$ and $\cdot:F\times V \to V$ satisfying certain axioms, as found for example on the wikipedia vector space page.

That these axioms are satisfied is easily verified. The first four amount to V being an abelian group, and that last four must be satisfied by F since it is subfield of R.

It also follows easily from these observations that if V is a vector space over a field K, then it is a vector space over any subfield F of K (i.e., the example above generalizes).

It is important to note, though, that V is NOT a finite dimensional vector space over F. The dimensional isn't even countable, in fact.
I think you're misinterpreting the question. Let's take a look at it:


  Let $V$ denote the set of all $m\times n$ matrices with real entries; so $V$ is a vector space over  $R$ by example 2 (note this was the general definition of matrices over $R$). Let $F$ be the field of rational numbers. Is $V$ a vector space over $F$ with the usual definitions of matrix addition and scalar multiplication?


Recall that a vector space is a set of indexed arrays (vectors) whose entries come from a field (scalars) that satisfy the vector space axioms. The field is required in the definition of the vector space just as much as the operations. It should not be taken for granted that addition is generally just to mean addition in the elementary sense. Certain fields can be paired with very odd operations that still satisfy the vector space axioms, and so we call the operation "addition" in the context of discussion of the vector space. Really, one might consider a vector space itself as a tuple of three elements($\mathbb{F},[\;]^{m \times n},+)$ with three entries that are first a field of scalars, an array structure to hold the scalars, and third an operation to combine the arrays that behaves like addition in certain ways. 

In the first sentence of the question, the author is establishing that the set of $m \times n$ arrays of real entries with Euclidean vector addition satisfies the axioms of a vector space. In the tuple notation, $(\mathbb{R},[\;]^{m \times n},+)$ is a vector space. 

In asking "Is V a vector space over F?", the author cannot possibly mean "Is the set of $m \times n$ matrices with real entries a vector space over $F$?" because to say "a vector space over $F$" means that the entries come from $F$, so the answer is a trivial no. It is trivial that $\mathbb{R}^{m \times n}$ is not a vector space over $F$, because by it's very definition it has as its field of scalars the real numbers. 

I don't believe that the author intended to ask a trivial question. Instead, the task is to determine if in the definition of $V$ one were to replace "real" with "rational", would the resulting $V$ given $(F,[\;]^{m \times n},+)$ also be a vector space. The question is non-trivial because $F$ is a subset of $\mathbb{R}$, so in this case if $(F,[\;]^{m \times n},+)$ is a vector space, then it is a subspace of $(\mathbb{R},[\;]^{m \times n},+)$ so the result can be demonstrated by showing that the set is closed under linear combinations.