Question

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)?

I'm a bit confused because we are narrowing our scalars selection from F to K. Although we are keeping the same Addition and Multiplication operations. Any thoughts about this?

Answer 1

A few thoughts...

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 " " <=> " " EE k in K : k != 0 ^^ a = kb`

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 `F "/" K` and a basis `B_2` of `V "/" F` then `V "/" K` has a basis `B = { b_1 b_2 : b_1 in B_1 ^^ b_2 in B_2 }`