Let V be a vector space over F. Explain why we can think of V as a vector space over K ( K subset of F)?

Question

fumanchuecf · Accepted Answer

Explanation:
First note that we require K to be a subfield of F - not just a subset. So it needs to be closed under addition, multiplication, additive inverse and multiplicative inverse of non-zero values.
Secondly note that F itself is a vector space over K. The addition of F satisfies the requirements for addition of vectors and multiplication by elements of K satisfies the requirements for scalar multiplication. The slightly painful thing is finding a basis. We can partition F into equivalence classes using the equivalence relation:

a - b \Leftrightarrow \exists k \in K : k \neq 0 \land a = k b

Then using the axiom of choice, a basis of F/K can be formed by choosing one representative from each equivalence class.
If we have a basis

B_{1}

of

\frac{F}{K}

and a basis

B_{2}

of

\frac{V}{F}

then

\frac{V}{K}

has a basis

B = {b_{1} b_{2} : b_{1} \in B_{1} \land b_{2} \in B_{2}}

Let V be a vector space over F. Explain why

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