# Mapping vector spaces over two different fields?

I was having linear algebra class and we have been discussing about a possible group homomorphism that might allow mapping between two vector spaces over two different fields

This is also an extension of this question

Suppose we have vector spaces $$V$$ and $$W$$ over some general field $$\mathbb{F}_1$$ and $$\mathbb{F}_2$$ and $$T$$ is a (linear) map from $$V$$ to $$W$$

In order to get around the issue of this vector space axiom becoming undefined because of c being in different fields

$$T(c\mathbf{x})=cT(\mathbf{x})$$

What's the issue in doing this (adapting the definition of group homomorphism, where there are two groups $$(G,@)$$ and $$(H,*)$$)?

$$\phi (a @b)=\phi(a)*\phi(b)$$

to the context of vector space (where the fields are defined as $$(\mathbb{F}_1,+,*)$$ and $$(\mathbb{F}_2,",@)$$)

$$T(c_\mathbb{F_1}*\mathbf{x})=T(c_\mathbb{F_1})@T(\mathbf{x})=c_\mathbb{F_2}@T(\mathbf{x})$$

(The two cs are different because they are elements of different fields)

It seems valid as long every element in $$\mathbb{F}_2$$ can be mapped from at least one in $$\mathbb{F}_1$$. What subtleties have we overlooked?

If this is valid is this still a linear algebra?

• Fields have a group structure given by addition and, if you take away $0$, one given by multiplication. Looks like you thought only of multiplication. What becomes out of $(a+b)v = av+bv$? (If you ask instead for a ring homomorphism, there will be only injective ones.)
– j.p.
Mar 16, 2015 at 13:47
• The notations $\mathbb F_1$ and $\mathbb F_2$ are rather misleading.... Mar 16, 2015 at 13:47
• $$T((a_\mathbb{F_1}+b_\mathbb{F_1})*\mathbf{x})=T(a_\mathbb{F_1}+b_\mathbb{F_1})@T(\mathbf{x})=(T(a_\mathbb{F_1})"T(b_\mathbb{F_1}))@T(\mathbf{x})=(a_\mathbb{F_2}"b_\mathbb{F_2})@T(\mathbf{x})=a_\mathbb{F_2}@T(\mathbf{x})"b_\mathbb{F_2}@T( \mathbf {x})$$will this work??? Mar 16, 2015 at 13:53
• @Secret Are $\;a_{F_1}, b_{F_1}\;$ scalars? Because if they are then you first have to tell us what can possibly be $\;T(a_{F_1}+b_{F_1})\;$ ... Mar 16, 2015 at 14:25
• Yes they are, and (for illustration) T can be a map from the reals (where $a_{\mathbb{F}_1}$ and $b_{\mathbb{F}_1}$ were in) to the complex numbers (where $a_{\mathbb{F}_2}$ and $b_{\mathbb{F}_2}$ were in) and at the same time maps elements in $V$ to elements in $W$. One example of T can be $$T: (a, \mathbf{v} ) \rightarrow (a+e^a i,\mathbf{w})$$ However the scalars can be from more abstract fields, they are not necessary have to be from the reals or complex respectively Mar 16, 2015 at 22:02

A map between two vector spaces over different fields cannot be linear (see this question), but can be semilinear . In this case there exists an homomorphism between the two fields $\phi:\mathbb{F}_1 \to \mathbb{F}_2$ that is also an homomorphism between the multiplicative groups of the fields.