0

Let $T: V \to W$ be a linear transformation, Is it necessary for V and W are vector space over same field?, if they are not over same field then what can we say about that?

Anju
  • 77
  • 8
  • 3
    Look at this: Can vector spaces over different fields be isomorphic? What is your definition of a linear transformation? Does it require V,W to be over the same field? – pem Jul 01 '21 at 18:25

1 Answers1

1

If $V$ and $W$ are vector spaces over fields of different characteristics, then requiring just $T(v_1)+T(v_2)=T(v_1+v_2)$ in general forces $T$ to map everything to $0_W$, which is not very interesting.

On the other hand, if the scalar fields have the same characteristic, then you can at least reinterpret $V$ and $W$ as vector spaces over the prime field, and speak about linear transformations between that. However, that ends up being the same concept as just homomorphisms between the additive groups of $V$ and $W$, and it's not necessarily enlightening to bring along the vector space machinery for that.

In the special case where one of the scalar fields is a subfield of the other -- say, one is $\mathbb R$ and the other is $\mathbb C$ -- it can make sense to speak of a linear transformation reinterpreting one of the spaces to be over the smaller field. However, beware that such a reinterpretation will generally change the nature of the vector space fairly radically: bases are not bases anymore, the dimension is different, etc. So your intuition about how linear transformation ought to behave might not be reliable anymore.

Troposphere
  • 7,158