I was wondering if someone could please shed some light on how the following two vector spaces are different. The examples are from Paul Halmos's "Finite Dimensional vector spaces" book and the author asks the reader to verify that they are different, but i am not so sure if i understand the two examples well enough to spot all the differences.
(1) Let $\mathbb C^1$ be the set of all complex numbers; if we interpret "$x + y$" and "$ax$" as ordinary complex numerical addition and multiplication, $\mathbb C^1$ becomes a complex vector space.
(2) If, in the set $\mathbb C$ of all complex numbers, addition is defined as usual and multiplication of a complex number by a real number is defined as usual, then $\mathbb C$ becomes a real vector space.
Is the difference between these two vector spaces examples only that in the first the scalar field is also the set $\mathbb C$ and in the second one the scalar field is $\mathbb R$ and hence only the multiplication and distributive properties would behave differently? What about dimensions of the two examples?
Any Help would be highly appreciated.
Cheers
Hardy
Best Answer
They are vector spaces over different fields. The first is a one-dimensional vector space over $\mathbb{C}$ ($\{ 1 \}$ is a basis) and the second is a two-dimensional vector space over $\mathbb{R}$ ($\{ 1, i \}$ is a basis).
This might have you wondering what exactly the difference is between the two perspectives. The point is that when you only consider real vector spaces, "multiplication by $i$" is not a well-defined operation. In other words, a complex vector space is precisely a real vector space together with a "multiplication by $i$" operation or, more formally, with a (real-)linear operator $J : V \to V$ such that $J^2 = -I$ (where $I$ is the identity).